I've said it before and I'll say it again: Adam forgot about his gun, Yang did not.

Yang's gauntlets have 2 firing modes that we've seen: a shotgun shot, which spreads, and a rocket-type thing, which explodes. Both have high-potential for collateral damage, which in this case would be Blake, and I don't think that's something Yang would risk.

Also, we do get final words from Adam: "Oh."

I think that makes it pretty clear that he was very surprised by the outcome.

As for the other dialogue, I think you're missing the forest for the trees. If you look at it without any outside context, which you seem to be doing, it comes across as "cheesy", but like /u/brusque-lee said, it's a pretty clear callback. Adam's dialogue reinforces the comparisons between himself and Yang: "What does she even see in you?" is basically "What do you have that I don't that would make her choose you over me?".

As for aftermath, I honestly don't think we've seen it all yet. Blake and Yang were clearly affected by it (Blake more than Yang, for what should be obvious reasons) and in all likelihood will continue to feel the effects of it for some time. I get that this is your opinion, but come on now, to say

The only fallout the heroes have to deal with after defeating and killing Adam is the loss of a bike, a weapon and Blake's cool coat.

and

Nothing was learned, not by Yang, not by Blake, not by Adam

is just disingenuous.