If Yogi decided that Dundee United was the next career step for him then I wouldn't begrudge him with a move there. In my opinion, he would leave us as a club legend for providing the club with:
Scottish Cup
European Football
Highest League Position Finish
Helping our top youth product secure a move to Celtic
Exciting brand of football
From the outlook, Dundee United doesn't look like an attractive job given the turmoil that Jackie McNamara suffered however I do think it's a job that could be a very attractive one too. Dundee United's youth academy is something with John Hughes's style of play and coaching, he could end up developing multiple Ryan Christies. There is obviously the apprehension of leaving Inverness just by looking at Terry Butcher for example. If he is given time and allowed persistence, I reckon John Hughes could be a great appointment for Dundee United and I'm seriously not surprised that they are considering him.
With that being said, I personally can't see Yogi wanting to leave Inverness and reading the article on BBC just backs it up for me. End of the day, the environment that Inverness provides John is something that works well for not only him but also us. One thing that he always hammers home is about grafting and team spirit, that's something I don't think would come so easily from the set go at Dundee United. Yogi could quite easily go to Tannadice and say we are going to play like Barcelona and they had a similar slump to what we had during our transitional period, they would be far less tolerant than we were.
One could argue, what's next for Yogi to achieve with Caley Thistle? In my opinion, he has much more that he can achieve with Inverness. Although, he has reached a high point with us before, there's no harm in aiming for higher and better things.
Going forward from here, I agree with other posters that he should be offered a new contract which in my opinion should have been given to him after winning the Scottish Cup. If we learned anything from the Butcher saga, is that stability is key.