You can write your own client for arbitrary services. You can run your own client on your own hardware, and you can host your client on someone else's hardware. Your client can render the content however it likes. You can have someone else do the work to render it like you like it and run that, locally or remotely.
There are a lot of ways to get to an "apparatus" that's analogous to both nitter and to a proxy for nytimes that ensure consuming public comments by other users doesn't suck.
As far as I know, there's no law requiring billboards to be seen by all drivers using the public road it sits beside, nor requiring all web clients to render ads. As far as I know, in most of the world there's no law saying users can't buy DNS or proxy or application firewall or ad blocking services from a third party -- services that can run between you and host, on your router, on your device at various OS levels, or as extensions to your browser.
On the contrary, there seem to be increasing numbers of laws requiring servers to deliver their content to arbitrary clients even when a firm makes its own servers, content, and preferred client devices and wishes to "protect the user's experience" or whatever the firm's justification might be.
Given a decade or two of marketing the "public square", other people's public posts, intended for public consumption, and expected to reach the public, picking that content as worthy of banning "useragents" should be shaky ground. Should be an interesting battleground for digital rights.
It's enough to wonder if a reason to go after the nitters is to establish misfortunate legal precedent using a target that can't defend itself?