TLDR: It's a very good thing.

Full answer: I've been a teacher previously (not private lessons, I had a work contract with my country's public body in charge of education), and I used to do this. It turned out to be a positive thing to do, for the following reasons:

• It allowed me to spot that one student that was having a really hard time with personal (medical) issues they did not dare to mention to the teaching team. Once they opened up about their problems, first to me and consequently to the rest of the pedagogical team, the situation clearly improved. And being aware of the support the teaching team had to provide released quite some pressure from their shoulders. I was warmly thanked by said student at the end of the year for opening the discussion about it.
• It also allowed me to spot the students who do not care. In particular, the ones that said they would make an effort, had a (very basic !) additional homework given by me, and did not do it. Which turned useful at the school council, in which we discussed what to do with the students who failed (allow them to retake the year ?).
• It overall gave me a good reputation as a teacher. Not because I was good above average at my subject, not because I was clear above average in my explanations, but because I cared about the students above average in my behavior. Or at least, that's what other teachers told me, and what quite some students told me at the end of the year.
• Lastly, if that's what's bothering you: Since I kept everything professional, i.e. I ensured I did nothing that could be interpreted as "the teacher getting into the students' private lives", I couldn't be reproached with anything. I was their teacher on a subject, they were clearly failing, because I cared about them I contacted them in an appropriate way (the email adresses they were supposed to use for teacher communication, fyi) to check that everything was alright, and if possible I offered them to discuss the matter during office hours at the school.

So it seems to me that you really care about our students, and that's a great thing. Offering to provide more efforts on your side so that your students succeed is a noble attitude for a teacher !

TLDR: It's a very good thing.

Full answer: I've been a teacher previously (not private lessons, I had a work contract with my country's public body in charge of education), and I used to do this. It turned out to be a positive thing to do, for the following reasons:

• It allowed me to spot that one student that was having a really hard time with personal (medical) issues they did not dare to mention to the teaching team. Once they opened up about their problems, first to me and consequently to the rest of the pedagogical team, the situation clearly improved. And being aware of the support the teaching team had to provide released quite some pressure from their shoulders. I was warmly thanked by said student at the end of the year for opening the discussion about it.
• It also allowed me to spot the students who do not care. In particular, the ones that said they would make an effort, had a (very basic !) additional homework given by me, and did not do it. Which turned useful at the school council, in which we discussed what to do with the students who failed (allow them to retake the year ?).
• It overall gave me a good reputation as a teacher. Not because I was good above average at my subject, not because I was clear above average in my explanations, but because I cared about the students above average in my behavior. Or at least, that's what other teachers told me, and what quite some students told me at the end of the year.
• Lastly, if that's what's bothering you: Since I kept everything professional, i.e. I ensured I did nothing that could be interpreted as "the teacher getting into the students' private lives", I couldn't be reproached with anything. I was their teacher on a subject, they were clearly failing, because I cared about them I contacted them in an appropriate way (the email adresses they were supposed to use for teacher communication, fyi) to check that everything was alright, and if possible I offered them to discuss the matter during office hours at the school.

So it seems to me that you really care about our students, and that's a great thing. Offering to provide more efforts on your side so that your students succeed is a noble attitude for a teacher !

Edit to clarify something following a relevant comment: I am not American, but French, and I taught in a public higher education system specific to my country (won't get into details about it, it'd be irrelevant). Thus, the local culture might have played a role in my whole experience with this email situation, and this cultural factor may affect the outcome for someone doing the same thing in a different setting.

TLDR: It's a very good thing.

Full answer: I've been a teacher previously (not private lessons, I had a work contract with my country's public body in charge of education), and I used to do this. It turned out to be a positive thing to do, for the following reasons:

• It allowed me to spot that one student that was having a really hard time with personal (medical) issues they did not dare to mention to the teaching team. Once they opened up about their problems, the situation clearly improved, also because being aware of the support the teaching team had to provide released quite some pressure from their shoulders. I was warmly thanked by said student at the end of the year for opening the discussion about it.
• It also allowed me to spot the students who do not care. In particular, the ones that said they would make an effort, had a (very basic !) additional homework given by me, and did not do it. Which turned useful at the school council, in which we discussed what to do with the students who failed (allow them to retake the year ?).
• It overall gave me a good reputation as a teacher. Not because I was good above average at my subject, not because I was clear above average in my explanations, but because I cared about the students above average in my behavior. Or at least, that's what other teachers told me, and what quite some students told me at the end of the year.
• Lastly, if that's what's bothering you: Since I kept everything professional, i.e. I ensured I did nothing that could be interpreted as "the teacher getting into the students' private lives", I couldn't be reproached with anything. I was their teacher on a subject, they were clearly failing, because I cared about them I contacted them in an appropriate way (the email adresses they were supposed to use for teacher communication, fyi) to check that everything was alright, and if possible I offered them to discuss the matter during office hours at the school.

So it seems to me that you really care about our students, and that's a great thing. Offering to provide more efforts on your side so that your students succeed is a noble attitude for a teacher !

TLDR: It's a very good thing.

Full answer: I've been a teacher previously (not private lessons, I had a work contract with my country's public body in charge of education), and I used to do this. It turned out to be a positive thing to do, for the following reasons:

• It allowed me to spot that one student that was having a really hard time with personal (medical) issues they did not dare to mention to the teaching team. Once they opened up about their problems, first to me and consequently to the rest of the pedagogical team, the situation clearly improved. And being aware of the support the teaching team had to provide released quite some pressure from their shoulders. I was warmly thanked by said student at the end of the year for opening the discussion about it.
• It also allowed me to spot the students who do not care. In particular, the ones that said they would make an effort, had a (very basic !) additional homework given by me, and did not do it. Which turned useful at the school council, in which we discussed what to do with the students who failed (allow them to retake the year ?).
• It overall gave me a good reputation as a teacher. Not because I was good above average at my subject, not because I was clear above average in my explanations, but because I cared about the students above average in my behavior. Or at least, that's what other teachers told me, and what quite some students told me at the end of the year.
• Lastly, if that's what's bothering you: Since I kept everything professional, i.e. I ensured I did nothing that could be interpreted as "the teacher getting into the students' private lives", I couldn't be reproached with anything. I was their teacher on a subject, they were clearly failing, because I cared about them I contacted them in an appropriate way (the email adresses they were supposed to use for teacher communication, fyi) to check that everything was alright, and if possible I offered them to discuss the matter during office hours at the school.

So it seems to me that you really care about our students, and that's a great thing. Offering to provide more efforts on your side so that your students succeed is a noble attitude for a teacher !