Like I said in my comment, I feel there's a few steps missing in your question. So, I'd suggest you first consider your short-term vs. your long-term goal again, then try to figure out if expectations are communicated and aligned correctly. Try to compromise, before resorting to kicking them out entirely.
When trying to manage a conflict like this, it's important to realize there are short-term and long-term goals. Your short-term goal is to get this player out of the group. The long-term goal is to stay friends outside of the gaming group, and also to be able to have no ill-feelings present when meeting up like you used to before corona.
I've encountered toxic players before. While the alliance leaders and me never succeeded in putting up the divergent boundary, the long-term goal of wanting to keep in contact with the good people I was gaming with overruled the immediate goal of setting the boundary and getting this player out of my hair. In such a case, you'll might have to grit your teeth and keep plodding on until you no longer need to game with them.
Given that you're good friends outside of the gaming and want that to remain, you might want to reconsider whether the short term solution of kicking out this player is worth possibly damaging long-term goals like staying friends without any ill feelings, especially if life is slowly returning to normal and you may be at the end of having to resort to gaming as a group activity. If the answer is that yes, this behavior needs handling no matter what effect is has on long-term goals, keep reading.
Your question and comments suggest you haven't tried a lot to tell this player that their behavior is bothering the entire group: You've told them their behavior isn't helpful. At the same time, that comment mentions that your current ranking is currently the highest it's ever been. That, to me, sounds contradictory: You're telling a person that's telling you how to improve (in a toxic way) that they're not helpful, but at the same time the team is improving.
I've had a few situations where things like this caused me or someone else to ignore what was said. It helps to make your words more clear: Instead of saying "Dude, you're not helpful", be clear about the effects of their behavior.
In this case, you can use I-statements: "When you say how poor my decision was, I feel demoralized and no longer have fun playing". When phrased like that, it will point out that the problem here isn't that the ranking isn't improving, but that the fun is being killed. If you're going to speak for the whole group, you can replace "I" with "we".
You might want to keep in mind some orientations to conflict:
There are three orientations to conflict: lose-lose, win-lose, and win-win. The lose-lose orientation is a type of conflict that tends to end negatively for all parties involved. A win-lose orientation results in one victorious party, usually at the expense of the other. The win-win orientation is one of the most essential concepts to conflict resolution. A win-win solution arrived at by integrative bargaining may be close to optimal for both parties. This approach engages in a cooperative approach rather than a competitive one.
Kicking someone out of the group would be a win-lose situation. You might first want to try to achieve a win-win situation instead. I've experienced those work a lot better if you want to keep good relationships with people, and your idea of compromising seems to suggest you're open to trying this.
You do mention you've come up with two possible compromises. A compromise requires two parties. Otherwise, you're just offering this person nothing more than a menu to choose from: Be kicked out entirely, play but mute, or talk but don't play. While it's always good to think about possible solutions, all of these seem very much focused only on what the rest of the group wants, and seem to be completely bypassing what the other person wants. Instead, find out what this person wants and try to compromise based on that too.
You're describing a player who is playing to win, and you mention they want decisions to benefit them as much as others. While you and the rest of the group seem to be there to hang out with friends, to do the group activity and have fun. Have you talked with them about their expectations for these gaming sessions? Perhaps the whole group could benefit to something that's similar to what's called a session 0 in role-playing games.
Talk about expectations, feelings and reasons for the behavior. Try to align and compromise. This is where the question from your title comes in: At this point, you can politely ask what it would take for them to no longer play with you all, or how you can make the experience the most optimized for everyone.
The compromise can be that they won't talk, or that they leave the gaming group and get to do other stuff with the group. It could be that they adapt a different style of coaching the team, that the team sets time aside after the game to talk about how to do better next time (and doesn't talk about that while gaming), it could be that instead of pushing you to do better, you ask them to joke and laugh about silly decisions. You could even perhaps pick a different activity or game. I hope the list illustrated the myriad of options. But they all require one thing: give this player a voice in the solution, in the compromise, instead of just telling them they're no longer welcome.
If you've tried to talk about expectations and to compromise and this didn't work, move to the next section.
Like I said before, kicking the player out of the group is a win-lose situation: there is one victorious party at the expense of the toxic player. It's going to be very, very hard to tell someone they're no longer welcome in the gaming group while expecting them to behave like nothing happened when you hang out at the bar.
If you've reached the point where this is needed, be honest. Also, warn first, if you haven't already. For the warning, you can revert back to the I-statements I linked before, and explain the reason using wording like that: "When you X, it makes us feel like we will be better off playing without you. We'd like to warn you that we're actually considering this, if nothing changes".
If that doesn't scare them straight, you will have to follow up on the warning. I've realized that if I warn people and don't follow up, they will take any further warnings or problems less serious. So, once warned, make sure you are all actually willing to follow up!
It takes a person that can take feedback very, very well for this to not create some sort of fall-out, even for just the warning. I've used it a few times, and it never went well in the moment. It took a lot of patience and making up after to get people to even realize they were cut off in the gentlest way possible. And that needed close friends, who were people that don't generally hold grudges. Less close friends or more grudgy people will usually move on and not look back.
I would advise against asking/telling this person to stop playing with you all, unless you're really ready for a last resort. If you decide it's needed and your relation is good enough to maybe pull this off (after all, you know your friends and the situation of this group the best), be prepared for some fall-out, and for needing something to make up.
