I've had good success booting Pi 3B and 3B+ computers from a variety of SSD and HDD models using a few different adapters.
The adapter cable I've had the best luck with is a USB 3.0 to SATA III cable by Eluteng, but unfortunately that adapter seems to have been recently replaced by a newer model that did not work as well when I tried it.
I also have an adapter cable by Sabrent
which I've tested with several SSD drives, and an external enclosure by ORICO
which I've tested with both SSD and HDD (names are links to Amazon USA).
My systems are powered by the official Raspberry Pi Universal Power Supply, and the drives are powered by the Pi USB port. Current Raspbian images are written directly to the SSD or HDD using Etcher, and my older 3B model has had the USB boot bit programmed.
As far as chipsets are concerned, it is, unfortunately more complicated than that. I have adapters with ASM1153 or JMS578 chips that do and don't work (same chips in different adapters). It's a combination of the chip and firmware in the adapter and the SATA drive, and how they interact with each other and the Pi boot-loader. It's possible to get combinations that don't work (no boot at all), partially work (cold boot fine, but fail on reboot), or work with some drives but not others (I have another adapter which works with my SP SSD but doesn't like my Samsung).
Yes, it is a bit of a crapshoot, but things have improved over the years, and compatibility is better now. The newer 3B+ has an improved boot-loader, and the Raspbian firmware has also fixed some issues. It's still not perfect, but it is much better than it used to be.
My mind is like a browser. 27 tabs are open, 9 aren't responding,
lots of pop-ups...and where is that annoying music coming from?