Transport stream is specifically meant for unstable connections vs Program streams used for DVDs with a nice steady data stream. Digital cable signals and other signals use transport streams specifically because they can resync if things do get out of sync.

But yes, working with TS feels kludgy. I haven't had to deal with them in over a decade, but there was one tool that made it all super easy, MP2TSME, that I hear is no longer available

The Internet you are referring to is not meant for low latency streaming. This is what it takes make it low latency. MPEGTS is proven stable and ubiquitous. If only it has less overhead.

Transport Stream is for everything, which includes the Internet.

I'm been using MPEG2 transport streams over the internet to broadcast programs to millions of people for over 15 years, and of course over private IP networks.

I'm almost certain you've seen some of that output