First I'm sorry you are here. There is a lot of helpful information for you in the healing library so definitely take a look there as well.
I agree with Cooley re red flags, but you know what those are as you are here. Then you said this:
I want to ask them to stay somewhere else for now, but I also don’t want that at the same time.
I am guessing (so please feel free to let me know if I misunderstand):
You want to ask your partner to leave because you know in your head and likely your heart too that you don't want or deserve the treatment they are giving you. You lack trust in them. You are being mistreated and kept in the dark. You are suspicious with good reason and the answers you are getting are not acceptable - not enough. Your partner has given you a million good reasons (likely more than you have written) to feel this way, and without change, you will likely continue to feel this way, so you know that asking them to leave will help YOU feel better eventually, if not today.
You don't want to ask your partner to leave because you are afraid they won't come back - that it will be over - and you don't want the person you fell in love with to be out of your life.
If I'm right about your whys, I can tell you the 100% biggest lesson I have learned from 5 years of dealing with infidelity and it's aftermath and so much great advice from others on this site who have been through it is this:
You can't control what someone else will do. You can't control the outcome when the outcome you desire involves another person.
You want to be with your partner but with certain conditions - that the lying, cheating, and hiding/shutting you out stops. You want intimacy. You want change from where your relationship is today. When you think about your partner staying with you I presume it's not the person they have become - you imagine them staying as you remember them before - when you were happy together. I presume if someone could show you a crystal ball into the future if you stayed with your partner and continued on the path you are on and the ball showed that nothing changed, that you were in the same situation 1, 2, 5 years from now that you are now, feeling the same way, you wouldn't want them to stay. I don't have a crystal ball...so you have to look at what you know - and what you know is now. Right now you are not okay with your relationship and how your partner is behaving. So do the only thing you can:
You can control how YOU behave. Therefore the best thing you can do for YOU is to tell your partner how you feel, what you need and want, and see how they respond and act accordingly. Only you can decide what deal breakers are for you and what you can live with, but I wish I had made these decisions sooner instead of staying in limbo-misery for years. If what your partner does is enough for you to stay together, then you stay. If they don't you ask them to leave and see what happens. Leaving isn't always permanent. Sometimes that push is what motivates a romantic partner to start working towards change - but sometimes it doesn't. It seems to me like unless you want to do some investigating - to "catch" them - then the time is now to speak up for yourself. It's hard (or it was for me) but it's worth it in the long run - no matter what the outcome of your current situation comes to, you will feel better having protected you - as you are the most important person in your life - don't forget that!
As to this:
I guess I want to understand what makes people think it’s a better idea to seek out other people instead of talking about their problems with their partner.
There are many "answers" but none are satisfactory as they don't make sense to those of us not standing in a wayward partner's shoes.