confiture.nvim VS compit

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.

It's not an exact solution for your problem but maybe it helps. I've built a small plugin which can be used to compile/run code called compit: https://github.com/PythonPizzaDE/compit