As a tourist, is this really the sort of thing one should expect?

i believe so. firstly - from regular people, not from those who make money on tourists. secondly - may be not in yerevan. (:
and it's not only for tourists. my friend a couple of months ago went to artsakh on his car with a girl. they had a car problem, and everybody around helped to solve it. when they parked at the small town which has no hotel, and decided to sleep in the car, somebody from the nearby building came and asked them to be his guests, drink tea, have food. they refused. then he came again. they refused and told him to leave them alone and let sleep. then he came third time, insisted that they sleep at his home. he left them in the house and left himself, to make them feel comfortable. appeared only in the morning.

Here in Denmark we're kind but notoriously reserved, so you might get help, but you're going to have to put some time into making actual friends.

Do you get many tourists there? Where are they typically from?

I don't know. now russia is under sanctions, they get less money, prefer to go to georgia and armenia more.
before that mostly us/europe tourists were here. i just judge of what i see on the streets, have no numbers.