"In a normal snow year, April 1 is considered the peak snowpack for the higher elevations. Memorial Day weekend (end of May) here is considered the start of the backpack season. Even then, a lot of the mountain passes are still closed."
I would say that Memorial Day weekend is a somewhat early start for backpack season, of course very much depending on the snowpack that year. PCT thru-hikers typically hike through there starting sometime in June, and in normal years it's considered somewhat tough/dangerous, with an ice axe being a very good idea.
In my year, perhaps a 'normal' to slightly 'high' year, I started through in early June and finish the JMT portion in late June. The issues weren't just snow (though there was lots of it), but creeks swollen by snowmelt at a lot tougher/more dangerous to cross than in normal times. And the transition zones could be a PITA, i.e., you drop back down from higher snowy areas (typically descending from a pass) down to lower non-snowy areas, and the transition in between would often be a sodden wet area of just-melted water running along the trail.
Another issue is postholing; one would typically try to do a "pass a day", and try to be getting down from the pass before afternoon sun and temperature softened the snow too much.
Still. If you have experience at backpacking in snow, it's not that bad. Consolidated spring snow isn't at all like hiking in actual winter, and there are less people out there (apart perhaps from a stream of thru-hikers headed north). You typically don't have to actually camp in snow, the mantra is "hike high and sleep low". And the bugs aren't out when the temps stay low enough, whereas they can get fierce shortly after (while there's still snow in the passes).