Just a quick response for now, will discuss more later:
Here is the problem, it has to remain closed source or people running the program could change a few lines of code and steal the bitcoin that everyone in the pool is working hard for. That is the only reason why it is closed source.
Wait, so the client controls the 'payout' or how could one's modification of the client on their side change how other contributors are paid out?

I think it should be possible to code it in a way that the client, which people run, is open source, so e.g. it can be compiled by users themselves and malware for example can be ruled out etc., but the server side software remains closed (similar to when mining with a pool where the pool software is closed and the client software like cgminer is open).
Or are you worried of someone modifying the client and distributing that version so that benefits go to them?
In that case, it should be the same way as when downloading Bitcoin core: you have a GitHub page where people are always pointed to, always recommended not to get it from anywhere else and on the GitHub they can either read and compile the source themselves or download and verify signed binaries that have been compiled for them by you.