Instead of making it achievement related, one thing they could do is to make each weave difficulty give a certain ammount of “illusions essence”, scaling up with difficulty, up to legend (or cata, because its a cata dlc kinda). This way there would be a little bit of grind to it, ofcourse balance it out so that it doesn’t take months to get.
The issues you mentioned about a group is also true outside of weaves. Premades can get their weekly quests way easier and faster. Someone who is alone or doesnt have alot of quickplay lobbies close to him, struggles aswell. The argument could be made that weaves are more difficult then a vanilla map and that bots are worse on them due to wind interactions.
The recruit issue will never be fixable. Also not really an issue in my eyes, but thats beside the point. You could make those “illusions essences” scale with difficulty which would encourage players to play higher difficulty, this could in turn also lead to toxicity, so its not ideal either. Another thing which would be more in the middle would be to make those “illusion essences” rewarded with time spent inside a map. So for example: i play a legend weave (it caps at 30, for example), so every minute you get 1 “illusion essence”, caps up to a certain amount and just rewards max ammount when completed faster. If this scales with difficulty, so recruit 10; veteran 15; etc, this could help alleviate the problem of having a team. Yes it would still be faster to do it with premade, but thats not really avoidable, unless you just wanna give every difficulty the same reward amount + use this time based system, which is a fair choice.
They could then even increase it further and tie those weave illusion rewards to ranked or leaderboards, or achievements in general.
You could finish in a certain spot on the leaderboard, e.g top 10, so 40 total rewards given out each season. which could be instead of unique rewards (as alot of vermintide players are againsts competition and unique rewards it seems) just 1 free weave illusion of choice.
They could make it so you get 1 free weave illusion of choice when completing a frame. So 1 for frame at 40 weaves, one at 80 and one at 120. This way everyone can get one. its not competition based as leaderboards would be and players who push more are rewarded more.
They could also just give them out just by doing general weave achievements. This way players who just play quickplay have a chance of getting a free weave illusion aswell, this would be limited and not be a choice, or maybe not, both could work.
So this way you can obtain weave illusions by playing normally, while also giving increased rewards every season or an extra incentive for players to play each season.
Another thing they could do is also give those “weave illusion essence” in normal vanilla games, as a reward on completing a map (like its with the athenor essence)(still needs to own the dlc to get them though), this way players who don’t wanna play weaves and don’t wanna feel forced can just play normal maps and if we combine that with the ranked/leaderboards/achievement system where you can choose 1 free weave illusion (or craft), this could increase incentive in weaves for players who do wanna get them a bit faster, and players who don’t enjoy them can also get them.
Just some ideas, its late, i might of missed something.