There is a lot of real anger about the changes here, and I think it's important to convey why this is. I'm going to try to do that from my own point of view: as a moderator, I'm not at all happy with the way things have been handled by the Powers That Be. (I think Rollout of new network site themes sums it up pretty well.)
We (the 'TeX-sx community') have always known that StackOverflow support the site for their own reasons, most likely as promotion for the 'main site' (StackOverflow itself). There's always been reasonable concern about the 'community building' statements from the Powers: at best these ideas are usually poorly handled. (For an example of this, see Community effort in fixing the double backslashes issue: many hours of effort by a small number of active users were needed to fix a problem introduced by the Power that could have been fixed easily by them, and which fundamentally broke 1000s of answers.)
It's quite clear that the TeX site falls into the 'Cadillac' set of network sites: we got set up several years ago and got a lot of design support when we graduated. So any design changes were always likely to be risky (detrimental) from a 'site improvement' sense: the old design wasn't broken. At the same time, I think most users can see that the network staff do have a complex task supporting the 'back end' of all of the sites, and can accept that some change will happen. What it needs to be is properly explained, proportionate and actually informed by user opinion.
The changes that have been made have been 'sold' as 'improvements for the network as a whole'. However, the reality is that there's nothing here for us or I suspect many other longer-established sites. Features that are useful for the main site or one or two other very large ones (SuperUser, ServerFault) don't make any positive contribution if you have under 100 questions per day. The Powers might imagine users using lots of network sites, but I doubt I'm alone in visiting basically just (this) one in any regular sense.
There are two fundamental problems with the changes. The first is the 'big picture': the three-column layout that uses up a lot of the screen space for 'filler'. I suspect that this makes sense if you are using a massive desktop monitor as one of a two- or three-screen setup, with your web browser maximised: in other words, the set up of a professional programmer using StackOverflow. I'm however not like that: I'm writing this on a 13" laptop with my browser window about 2/3 of the screen width. (I've disabled the left sidebar, too, precisely because the design doesn't work.) The fact that this change has been made and not rolled back quickly shows a total lack of real engagement with users. Telling us 'this will help you' is a non-answer to the criticism: the new 'features' we are being told we'll get are totally useless, at least as currently described.
The second aspect is the smaller changes to design. There's a suggestion that these are needed to bring the 'back end' of sites into line, but they feel arbitrary and poorly implemented. For example, there are no obvious technical reasons why we have to loose custom graphics for voting buttons. I can think of only two reasons for this. The first is that StackOverflow wants to make the sites look more similar for advertising reasons. The second is that newer sites won't be getting custom graphics for resource reasons, so those of us who already have them loose out as 'it has to be fair'. Crucially, neither of these is anything to do with the community on this site. If there are things that have to be done for management reasons, including balancing resource usage, then tell people: the network is full of people with a technical background, for goodness sake. (See https://meta.stackexchange.com/a/312404 for another place that things feel entirely arbitrary, from a non-TeX site.)
A boycott has been raised, and some people will doubtless walk away, primarily because of the completely tone-deaf approach of the Powers That Be here. (It shows yet again that all of the 'community building' stuff is so much guff.) I'm not about to walk away, but as a community we are lucky that there is TeX beyond TeX-sx: we do have an international user group, people with server experience, etc. So if some sense of listening doesn't happen, I would not be surprised if a truly 'community owned' Q&A were to be established. I'd certainly register for such a site.
So what should StackOverflow do? I think you need to roll back the changes (network-wide), and think about what message you are actually trying to put across. If you really believe in community building, then listen to users. Moreover, if there are corporate reasons that some changes have to happen, then front-up and say that: we can understand 'The company has to balance its books, and this will help by ....'. The design changes should have been made step-wise over a period, not all in one go with both layout and resources changing. The key is to actually show you are listening, and to recognise that each site is independent (see for example https://meta.stackexchange.com/a/312406 pointing more-or-less exactly that for another site).