Diversity in licenses is a sign of cooperation.
You cant just mix licenses, because licenses disallow/allow activities.
You can't have both a no-ideological community and "no gpl welcome". Either you welcome everyone and focus on getting code to work, or you exclude based on licenses and reinvent the wheel. The question you need to ask is what is important for you. If the most important thing for the project is the license choice, then be upfront about the ideological choice that such goal is. If the goal is to get working code that will use any software you can legally use, then a more pragmatic approach is more suitable.
I personally don't run operative system which focus more on the license rather than getting things done. My work also don't pay me for license choices, but they do for code that works and create value for the customers. I could put agpl code on the servers and you know what, no one would object.
You can legaly include apache in GPL software.
Following your theory, one would either have to break GPL and risk legal problems or exclude close source users of his work.
But I digress. There is a clear bright line between writing code to solve a problem, and rewriting code because the license don't let you create property in the form of closed code. One is about pragmatic problem solving, and one is about ideology. One allows for code reuse, one is limited to my-way-is-the-only-way.
If you live in an ideology that demand that everything you do must be made into property owned by the company and only secondary solves problem for a customer, you will be slower than a companies which more pragmatically focused on fixing problems for customers. Don't blame others if you are self-imposing such restrictions on yourself.
It absolutely does. It fixes problem of people who work on close source or otherwise something incompatible is gpl. You either dont count them as people or as problems and neither makes sense. I used open source inside closed source. So yep, if I am going to five away, I will give to the same pool. Generous people gave me something, so I am giving back.
A library that I can not use does not solve any problem. That is not pragmatic, just something with limited use. Which is fine, if you feel strongly about your work use being restricted to some people only. I don't.
The moment when your logic twists gpl into literally "the only pragmatic license" is the moment when I wonder whether you are even honest in your arguments.