Sometimes I have to wonder what the point of even having levels 1 and 2 in the game is since it's almost practically a given that a DM will start off their players at level 3 minimum. I guess it's more newbie-friendly to have the option to start players off with less stuff to have to learn right off the bat, but is it really that much more at level 3? Like you said, most classes barely have anything useful or fun those first couple of levels, even to the extent of not even coming close to how they play once they have all the core components of their toolkit. A lot of the time, your options are to do a basic attack and not much else, to say nothing of things you can do outside of combat. And any adventures you go on have to be of the absolute lowest stakes to avoid dying in the first session (insert debate about the morality of throwing mudfarmers into the meat grinder here). Even then, it's not impossible to get knocked down that early; I remember one level 1 character nearly dying to a single giant rat in our first combat encounter.
I suppose it's best to have an established low baseline that can be worked from instead of having a bunch of stuff by default, but it's funny to think that the number of players that have played both very high and very low level characters might not be too far apart from each other. Generous of your DM to let you all get max HP for that level, though.