However, as A-Team remarked earlier this melt season, the time of peak solar activity does not overlap well with the time of minimum ice.
Why would you expect it to? Even in a world where sunlight is the main cause of ice melt, then the peak solar activity would not coincide with the point of minimum ice but with the peak point of ice
loss - i.e. the steepest part of the yearly ice loss curve during the melt season.
As it happens, there's a little bit of extra lag beyond that, because it takes a while for various short-term summer feedbacks (snow melt lowering albedo, melt ponds and leads likewise, and the sheer time taken to melt through the ice layer) to kick in. Not much of a lag though - the point of fastest melt rate typically occurs within 2-4 weeks after the solstice.
The inflection point between melting and freezing (i.e. the annual minimum) happens at or around the equinox, more or less exactly as you'd expect from that being the point where solar input (or rather radiation balance) starts to turn negative.
If you approximate ice extent as a yearly sine wave, it's offset by 1/4 of a cycle from the solar input cycle, exactly as you'd expect if insolation was the main factor affecting melt and freeze-up. Sure, there are other interesting dynamics from warm air and warm water intrusion - and we discuss them to death every year - but they're second order effects.