MPEG-2 TS is a container. H.264 is a coding specification. They are totally different things.

One can find MPEG-2 TS in surprising places (see: DOCSIS encapsulating Ethernet frames into TS packets).

If I had to guess why MPEG-2 TS, it'd probably be the for the fact it's a well-supported streaming format in both hardware and software. If you tried using QuickTime or MPEG-4 containers, you'd have to rely on hacks like ensuring the moov atom preceeds mdat.

Matroska may be worth considering (especially the subset used by WebM to make it stream-friendly and quicker to seek), but no idea how widespread hardware support is for (de)muxing that.

MPEG-TS is used to contain h264 chunks for HLS. MPEG-DASH and the new CMAF standard use fMP4 containers instead. My personal take is that Media over QUIC (MoQ) should support both.

No, but under it ;)