Seeing it as a kid, that was a “holy crap!” scene - and I _got_it_ instantly: Han knew he’d be dead if he didn’t stop Greedo right then, and there was only one way to do it, which he was sneakily preparing to do under the table. There was no pussy-foot reason for him to wait to get shot at (why would Greedo shoot him _there_? when obvious Han was wanted _alive_?), in all likelihood getting killed in the process before returning fire on politically correct terms. YES it was OK for Leia to later fall for a man who would do what needed to be done.
I just thought that the other guy should have pulled his weapon and then Han could have shot him from under the table.
The rest of the bar ignoring it and going back to minding their own business quickly drove home a bigger point right afterwards, though, that death was typical in that bar and it was as dangerous a place as obi-wan said
He who shoots first gets the princess ... is that something like the early bird getting the worm?
Need to practice up on my Weaver stance, I guess.