I don't see any reason to expect it would. Summer months, for example, already apply a strong negative pressure on the flu virus (that is, we have a flu season

I don't see any reason to expect it would. Summer months, for example, already apply a strong negative pressure on the flu virus (that is, we have a flu season), yet we have not seen a year-round flu. Diseases always face selection pressure to spread more easily and rapidly. When that selection pressure is caused by broad factors like "people aren't congregating together," it's hard to imagine how any simple evolution could overcome that barrier. Covid itself is instructive here. Some of the variants of concern are more transmissible than the originally-distributed variant as of March of last year, but those mutations don't uniquely overcome masks or social distancing. They would have always result in a more transmissible virus (because SARS-CoV-2 appears to be a zoonotic virus not originally perfectly-adapted to humans), and we are seeing them now because each infected person is a new roll of the mutation dice. Applying that lesson to the flu, the less prevalent the virus the less likely we are to see variations, not more. Alas, for influenza itself we always have the asterisk that we regularly see new variations cross to humans from animal reservoirs, so we're unlikely to eliminate it through only (nonpharmaceutical) human-health measures. (It'd be nice to get a universal flu vaccine, though.)