According to the Vintage Television Museum near Columbus, Ohio, the last company in the US to be able to do this closed in 2010, and the last one remaining in Europe closed in France in 2013. (I myself don't know if there are any in some other corner of the world.)
The museum did succeed in getting a bunch of the repair equipment from the shop in France, and one person involved was even trained there, but it's been a very long process.
Currently, the equipment seems to be in Maryland in the care of a person named Nick Williams. The last update I can find from him[2] is a few years old, and expressed concern about the war that had recently begun in Ukraine affecting the supply of electron guns.
tl;dr, it may still be possible to repair some aspects of some CRTs, and doing so is apparently not a completely-lost art -- yet.
[1] https://www.earlytelevision.org/crt_project.html
[2] https://www.earlytelevision.org/nick_report_5-1-2022.html
Repairing the tvs, sure, but i find it hard to believe there were repair shops for the issue parent was mentioning.
I can well imagine that it’s gotten expensive finding a quality one (eg trinitron) of reasonable size.
That said, yeah, in the special case of an HDMI-driven CRT that was specifically designed with ultra low latency in mind, you could buffer way less than a frame— though I imagine you'd probably want to buffer at least a line at a time just for sanity with the timing of driving the electron gun. And obviously this would depend on the HDMI picture resolution exactly matching that of the CRT.
LLMs probably don't know enough about them to be useful in this discussion. Classic Google Search would be better. Yours fixating on pixels shows that.