First and foremost. Most people’s devices are not powerful enough to make any money mining any cryptocurrency.

That depends on the currency. Miners working on something like bitcoin have to compete with massive corporations who construct special ASIC chips that have vastly higher hash rates, making it impossible for anyone who isn't using such a chip to create any profit. Some currencies, like Monero (XMR) use special hash functions designed is such a way that custom mining hardware for them cant be constructed, and so mining can only happen on cpus. Because of this, its actually possible to make money mining Monero. There's a benchmark for the hash rates of various cpus, and here's a link to convert that to projected profits. My PC can calculate 3500 hashes per second, and this translates into USD 0.0045 per hour. That might not seem like much, but if you add together all the money that would be produced by all the players in this game, it would become substantial.

Also a cryptominer is not “free real state” it chugs the computer. The user would have a terrible experience trying to do anything with a cryptominer on the background.

That all depends on how many threads the miner is running. I honestly wouldn't be able tell if I'm running 2 threads of a Monero miner on my PC unless I look at my task manager - its really not that intensive at that level. Now of course, if it was using the full power of my cpu, I would be a lot hard to do any other task on it, but it doesn't need to run at full power like that.

And finally, there are many free software out there. Not everything is to be monetized. Some things should just be free. I have done plenty of free things for others to enjoy, it’s not the end of the world, quite the opposite is quite rewarding.

I agree and this game costs something to run, all the server costs and the API costs, and there needs to be some way of paying for that. I suppose it could become free if it was sustained with donations, and in that case having the option to run a miner like this on the client could be a way to add to that, giving the user the option to donate their computing power if they wanted to.