Don’t most home HVAC systems already filter the air blown through them? I know I replace mine every so often. If they do, what does a separate “air purifier” buy you?
Not sure how much you'd gain, but one thing to note here is that I believe not all HVAC systems have an option of running just the blower w/o also heating the air.
The only furnace I've seen without that option was from the 1950s (probably replaced by now for old age), but I'm sure more exist. It wouldn't be hard from someone who knows power to wire up the fan to run all the time though.
Furnace is one thing, the wiring in place is another I guess. My house is from the '50s, and the furnace was already replaced, but we have not laid new wiring. I think I need to look into that before the next wildfire season starts...