Well, at least they tried to explain why Thawne wasn't retroactively erased from time, although the explanation doesn't make a damned bit of sense. Nor did the randomness of the alteration to the timeline affecting Cisco and nobody else. If this is the kind of temporal "logic" the Arrowverse producers have devised, we'd better not expect a lot of coherence from Legends of Tomorrow. Still, it's a good way to establish Thawne as a potentially recurring adversary and to explore his enmity with the Flash more fully.
Kinda sucks that the only reason Thawne became a villain is because he learned that history recorded that he became a villain. Although, really, that sounds more like he was predisposed to villainy anyway and let "destiny" be his excuse.
And why wouldn't facial recognition turn up someone who had the same face as Jay but a different name? Isn't that exactly the situation where it would be most useful?
The bit about Iris and Wally saying their goodbyes to their mother rang true to me. I went through a similar conversation when my father was in hospice care. He wasn't conscious, and I didn't see much point in going in to talk to him, but the hospice worker convinced me to do it and get some closure, and I did feel better afterward. I was struck by how cliched it all felt, like something I'd seen in a million stories, but then, some things become cliches because they're basic truths that apply most everywhere. (David Gerrold once wrote, "Given enough time, all the world's great wisdom will be expressed as cliches.")
That was a really dirty trick by Patty at the end there. In fact, given that Barry is a police employee, calling in a false crime report to him is probably illegal.
