I can only assume that the pressure vessel committees disagree with that assessment. Note that A36 does use lower allowable stresses, and does have different treatment for notch toughness than A516-70. So while a mill might downgrade a batch of A516-70 into A36, it wouldn't be designed like it was still A516-70, either. For that matter, a mill could downgrade some higher-strength plate into A516-70, but that wouldn't be a reason to avoid A516-70 in use. In the applications I'm familiar with, I'm not aware of any quality problems with A36 specifically. I've seen similar statements made on US-vs-domestic plate, and haven't really seen a trend there, either.
As to which is actually most common, I think that would have to depend on the experience of the user. I recall talking to one vessel fabricator that made virtually all their vessels from pipe. Or if you're using anything much over 1/2" or 3/8", it's going to be more economical to use A516-70 or some other material anyway, because of the higher allowables. F&D heads are commonly stocked in A516-70, and may or may not be available in A36. I'll bet half of all the ASME vessels in existence are air compressor tanks or propane tanks, and that would certainly skew the statistics to whatever those people use.
By the way, I recently saw a little aluminum air tank for sale (pressurize it, then use it to air up tires- maybe 5 or 10 gallons). Boldly stenciled on it was "Do Not Drop". That, to me, is scary.