Non profit does not mean that it do not have to pay taxes on the money it manipulates.

Non profit orgs receive services from the state and they have a cost to fund social security and such.

Other non profits which do not have a very nice situation can receive help from the state, so why not share the bits of success of Lichess to everyone else?

It's the norm in France for employers to pay another euro of tax for each euro their employees earn.

I assume with "tax" you mean "tax + health insurance + unemployment insurance + retirement savings etc"?

Yes and it was a mistake in my opinion. It make employees feel that the company is paying for these benefits and it makes the company believe that they are spending money for things they should not (I've heard so many CEOs complaining about having to pay "all those taxes for their employees"). I prefer a way where the employee is responsible for all costs with a deduction at the source to make sure you don't get nasty surprises at the end of the year. In term of numbers that's all the same, not in term of psychology.

Incentives and human responses are endlessly weird and interesting.

Sure

Why should non-profit be tax exempt at all?

Because the state was never about making things easier for people.