This is going to be a hard ride.
You don't just have a player pair of clashing styles. You need a few stern words that might sound harsh, but I have learned over years of GMming, that sometimes you need to hear the truth that you messed up. I might be loud for the next paragraphs, it might read harsh, but I will do my best to put together a way out in the end, or at least give you a pointer on what I would do. So bear with me, ok? So..
You did not live up to your goals.
You start your tale with this:
I would have shut this down, but the rest of the party seemed OK with the idea,[...]
And then, right away, let's stop here for a moment. You did not want it? But you are the GM. You are the one-and-only final arbiter in the campaign. You could have stopped it, and yet you didn't. Remember, the DMG tells you that You are always right for a reason. Yes, we all know that with great power comes great responsibility1. And your responsibility is to enforce the one and only rule that is written down as "Rule 0" at times: make sure everybody has fun. And sometimes, that requires you to say no.
You say:
I'm worried that, since Archel's making the campaign too easy, [...]
That can only happen if you always say yes and always allow whatever Archel suggests. But that is not how you run a game of Dungeons and Dragons. That's how you play a game of powered-by-the-apocalypse or a freeform, where every player is allowed and encouraged to bring in new aspects and create the world together. It's a behavior that is at home in games like blades-in-the-dark, which has its own mechanics to add such stuff at the cost of stress, or exalted-3e, where making up stuff to detail the situation gives a bonus on the related roll. But the game you play is more hardline on what it allows to players, and it is much less freeform.
In Dungeons and Dragons, it is your task to make the world and make rulings so that everybody has fun. It is your solemn duty and power to shut down things if needed to tell a good story.
Is it a good story and everybody has fun when everybody does add some details? Great, play it loose. You're not playing by the book, but the goal is to have fun anyway.
But here you have two players clashing: One player wants to play a much more freeform game and a character that appears to be somewhat similar to the main character from "A Trekkie's Tale"2 (Lt. Mary Sue): She knows everything and everyone and has a solution up her sleeve... And that can be fine until it impedes the fun of others. And it clearly does, as Eistera feels bothered and wants to shut down Archel's additions for it.
And let me be frank here: The problem is not Archel and Eistera, or their clashing styles, the problem is, that you as a GM have to draw a line, and enforce it. It is your job as a GM to say: Stop, no, you can not do that. It is your job to make sure everybody can have fun. And everybody includes you. You say:
I also don't want to confront Archel head-on, as the other players [But Eistera] don't have a problem with him.
You forget about one person: Yourself. You have a problem with the situation. You have come here to ask for advice on how to handle it. So let's grab a hold of the situation and see what we can do!
A few pages from my experience
I am somewhat seasoned, I'd say. I mean, I play since more than a decade, I have played with some bedrock people that started with 1st or 2nd editions, I have played at cons and in RPG clubs. I have not seen everything, but I have seen a lot. And some of that seems applicable to make you become a better GM and aim to achieve your goal:
Communication of expectations and rules is key
You need to step up and state clearly what is and isn't acceptable at the table. You should come up with simple table rules that are easy to enforce. Best, this is done during a Session 0, but I have introduced table rules later with some success. The biggest point is to talk about it. In effect, you want to form a Social Contract at the table. A set of rules everybody is bound by. I think you can feel inspired by the very basic rule that was introduced by Traveller:
Obnoxious, obstreperous or rude behavior is not conducive to the enjoyment of play. Loud, disruptive players merely irritate everyone concerned. Do not engage in such behavior yourself, and discourage it in others.3
For your specific situation, let me suggest a rule that I use in a game where experienced players are allowed a wide latitude of bullshitting stuff and situations up, usually to flavor interactions:
- If you invent or make up a fact about the world, it can't be a painless, free, and perfect solution to a problem you face. The GM has the right to veto such suggestions wholesale and alter them as required.
That sets expectations and indicates that you as a GM can and will enforce consistency with the world. Other typical rules I have and encounter - at times unspoken and which might help you steer situations, and most of them are basically safety-tools:
- Don't talk over one another.
- If you are overwhelmed, make the GM aware with the X-Card/X-gesture/timeout-gesture or loudly say Stop.
- Keep off-topic to a minimum.
- No Rules Lawyering.
Learn to say No at times
A big part of the situation came from you not saying no at all. Or at least letting things happen as suggested with no negative consequences at all.
But the advancement of the plot always has a price.
If you want to get into the town by pulling Vitamin Connection, there is a price to pay. Things backfire (remember, you can demand charisma checks to see if your bluff or story took hold!), and even the best shoddy plan has holes, usually big enough to walk an elephant through. Think of it as an escalation strategy:
- If you disagree with the suggestion fundamentally, just tell them: "This does not work", or "no".
- If you agree to the basic premise of a suggestion, but not all of it: Say yes, but... and let them fail forward.
- If you agree, but see problems they might need to work around let them roll for success.
- If you agree wholeheartedly and there is nothing to lose, say yes.
Have the sheets less fungible
It might help to have the players pin down their backstories a little more.
For such, I suggest something I encountered in a game: we wrote down places where our characters have been on the continent at specific points of the timeline, or what they did then. This later allowed the GM to not just feed information to characters that were at a specific event as required, it also allowed them to interconnect characters. Not every field was mandatory, but the more you filled in, the less fungible other fields became. Some stuff was filled in during the game as players decided, but the very idea of writing down things for later referencing helped us to constrain the backstories.
- Stan Lee: Amazing Fantasy #15 (1962).
- Paula Smith: A Trekkie's Tale, In: Paula Smith & Sharon Ferraro: Menagerie (1973).
- An Introduction to Traveller (1981)