No argument about the noise of the box fan-- although this can vary widely depending on the fan.
I also don't know about the differences between carbon filters and hepa filters, and what they filter out.
But I'll note that the cleaning properties of any specific filter is very directly related to the 'dirty' air that the fan actually moves through the filter, and the filter needing to be replaced might often be an indicator that the fan/filter is doing a good job of actually filtering dirty air. If a small purifier has filters that last a year, there's a very good chance that it's filtering far less air than a larger fan would filter.
Another concern is the effectiveness of fan baffling or shroud, which keeps untreated air being fed to the input side, instead of allowing the exhaust-side air to be pulled back around to the input side.
With forced air central HVAC, the isolation of input side and exhaust side is typically very good. Simply using higher grade filters in the HVAC system might be the most effective method of filtering dust, even if (and perhaps especially if) the filters need to be replaced frequently.
Re the noise, she said it became "white noise" to her after the first few nights but no doubt that a box fan is probably louder. She ran it on high for a bit (1 hr.?) before bed time then low all night, I suppose an air purifier on low setting is going to be quieter.
I think you need to look at the total filtering area. The purifier filters are quite small but usually thicker than a 20" x20" x1". Another factor is how densely pleated it is. I have bought HEPA filters for under $12 in qty. (8 or 12?, our air handler uses 2 at a time) but we stopped using them since we saw no difference in dust in the house and airflow was reduced even with new filters. The strain on the fan motor was marked (30% inc. draw), I didn't like the idea of the inc. load re motor life on the old system that wasn't designed for anything other than freer flowing filters.
Her setup lasted her a year, she replaces the filters every spring. It is two filters, a cheap standard "spun" filter then the HEPA filter. Her bedroom is not large and she generally kept the bedroom windows closed at night. Perhaps having a "prefilter" in the form of the $1 spun filter helps the usable life of the HEPA filter.
This setup won't do a thing for odors like carbon filters are designed to do. I've seen carbon impregnated furnace filters but doubt they'd be nearly as effective as the multifilter air cleaners that have separate carbon filters or ionization sections.
ETA - a quick look on amazon shows 20x20x1 HEPA filters in the $8-$12 range depending on brand, # in multipack & MERV #. Noticed they also have 20x20x2" ones also (@ 2x price naturally). Maybe these aren't the top of the line filters realLexusl21 was referring to though, IDK.