confiture.nvim VS nostalgic-term.nvim

This is also not a direct response to your question as I'm not familiar with asycntasks.nvim, but having a build and run functionality is probably the main reason why I wrote confiture.nvim, which is another task launcher plugin (that also aims at being simpler than overseer). The main limitation with the implementation of the build and run of confiture is that the build phase will not be asynchronous. It is sufficient for my workflow though -- if I know the build command will be long, I usually launch it on its own first, which can be asynchronous using confiture (with vim-dispatch under the hood).

It's can also be used from vim to launch custom commands (as is done for example in my other plugin confiture.nvim)

It doesn't answer your question but this is how I do what you described : https://github.com/romainchapou/confiture.nvim

I'm mostly fine with language specific configuration, which I find not too hard to manage in my init.vim, but I rely a lot on project specific commands for building and running projects, so I'll do a shameless plug for my plugin, confiture.nvim, which aims to provide a simple solution for this.

This looks cool! In case you might be interested, I'm the author of a similar plugin, but which only does the mode saving in terminal buffers as in Vim8 (and some other small things), nostalgic-term.nvim, and I did manage to make it work with ... type mappings and switching windows using the mouse. The main trick I used was to set the saved mode for a terminal to normal mode only if the user spends more than a dozen milliseconds in a terminal buffer in normal mode (see this bit of code in nostalgic-term). This means a ... mapping won't count as switching to terminal normal mode, but actually switching to it with will count.