So I've had contact with Mathworks support and the issue is fixed!
The problem was that we installed our targets by moving them to the matlabroot/rtw/c folder. But this not allowed/supported:
"In R2020a, we introduced the new Simulink Toolstrip. And it was decided that when launching the "Run on custom hardware" app, it checks the location of the .tlc file specified in the model configuration. If the .tlc file is under matlabroot, then only certain MathWorks provided targets are supported, but not custom targets."
So the trivial fix is to put custom targets in another folder but NOT under matlabroot. Then make sure this path is added to matlabpath and save the matlabpath.
Now, when opening a model with your custom target, the hardware toolstrip immediately pops up and you're able to use external mode!