The shortest answer is "For sure, drones help with staying in tune."
(A short version of the longer answer is "Is staying in one key always a good thing?")
A drone can be very helpful for practicing intonation (being in tune). I use it often myself when practicing violin: I have a metronome that can also play any note, and e.g. set it to an E and then play some E scales or arpeggios. I can tune any E notes I play to sound exactly the same, and for the other notes, I can still listen to their difference. You start to get a feeling for how an A sounds against an E, and can get fine-tune that distance.
And yes, it also helps not just with fine-tuning (like "is that exactly an E or is it 1/10 of a step away?") but with much bigger issues of drifting out of pitch (like "you aren't in E anymore; you went straight through the keys of D and C and sank all the way into B"). It might not always save every situation, but it helps provide a sense of reference. It's a great starting point if someone has problems with matching pitch. (Even better is to play the melody on an instrument like piano while singing it, to make sure you match exactly, but that doesn't help much if you're making up your own music.)
By the way, this is used in rap pretty often. Consider Jay-Z here:
The music is pretty much sitting on one single chord. And although we think of rap as "spoken," it's often quite pitched. Jay-Z spends a lot of time with his voice around certain frequencies. Listen to "So you ain't gotta feel no way about Jay so long"; it's practically sung. (To get more into the particulars, he's not even spending most of his time on the "main note" of this chord, but one note lower, giving it a bit of tension that feels like it's not at rest but moving forward.)
Now, there might be some times that a drone doesn't work with some musical structures. True, the vast majority of songs spend their whole time in only one key. But some don't; some have the chorus in a different key (or mode) than the verse, or even go to some distant key for a bridge. I think of the song "Hummingbird" from Spiderman: Across the Spiderverse:
The whole song as a rather "unsettled" sense of key, "hovering" like a hummingbird around notes near to the tonic without ever "alighting" on it. But when the lyrics get to the title word, "Hummingbird," we get a break (hook? chorus?) with a big sense of harmonic change. (Like I say, it's a hard song to analyze and is intentionally harmonically "fuzzy," but I think it might be tonicizing the dominant.)
Or some songs simply modulate (think "Mack the Knife," which just repeats the same verse material but shifts the key up by a step each time). When I practice music that changes key along with a drone, sometimes I have to pause and change the drone to a different pitch for a section.
Honestly, for your own composition, this is something you don't have to worry about at first, and are under no obligation to worry about ever. Definitely, knowing which key you're in and staying in it confidently is the first skill to master, and is a requirement before you start taking the melody for a tour of other keys. And don't make the mistake of thinking that music that does more complicated things is necessarily "better." Simplicity can be a virtue, and growth in complexity is not necessarily evolution. But if, somewhere down the road, you start getting bored with your composition within a single key, try shaking it up by switching to a different tonal center for a section of the song.