> In service of the BTC network, which has what utility?

The safest safe haven asset. A global currency of fixed supply controlled by computer algorithms.

> There are quite a number of criminal or at least grey market uses for BTC.

Criminals the world over prefer cash. Should we ban cash?

> But what are the legitimate uses that are not as well or better served by the traditional payment networks?

A programmable, internet-native inflation-proof store of value with transparent fully-auditable supply metrics. Or to quote Elon Musk, “simply a less dumb form of liquidity than cash”.

It's less a question of if Bitcoin has utility and more about the proportionality of its utility to resource usage. The problem is that there's always incentives to increase the amount of processing power being thrown at Bitcoin but the utility of Bitcoin is independent of that. You can change the amount of processing power in the network without affecting the rate of transaction processing, rate of minting, or the perceived value of Bitcoin.

Bitcoin's true use case hasn't showed itself yet because it is uniquely suited to help us in case of a select few low probability and extreme scenarios, aka black swan events which we can't predict, but will probably happen.

Operating data centers makes the data center owners more wealthy. All in service of the data center network, which has what utility?

There are quite a number of criminal and grey market uses for data centers.

What are the legitimate uses of data centers that are not served by traditional pen & paper?