I don't think it's impossible to mix the two successfully, but I think it varies greatly by what the project is. Some things monetize very naturally, other things would completely lose their appeal if you tried to contort them into a product that people pay for.

Example: I did a project for a hackathon at UT when I was in college. The idea was that in this day and age it's still hard to get a file or snippet of text from one device to another unless they share some kind of account sign-in or something. People still email things to themselves.

So I made a website called "Catch" where you can upload a file or a text snippet, get a throwaway 6-digit code with which you can download the thing from any web browser, and after ~15 minutes the artifact is deleted from the server. The whole appeal of it was having absolutely minimum friction.

When we presented our project one of the first things they asked was, "how will you monetize it?". It was something where any sort of sign-up or payment step completely defeated the purpose, so the answer was, "we won't".

But I don't think that's the case for every fun-project. Especially if you take a freemium approach or show ads.