As written, no.
The variant rule does not include progression.
As the other answer didn't quote the whole section, I'm not going to either, but it doesn't mention progression. It mentions rolling for stats and how to give them equipment, but there's no mention of them gaining experience or accruing class levels. While it doesn't explicitly say they can't gain experience, it also doesn't explicitly say they can. My reading is that, by default, even with the variant rule, they would not progress.
Houserules are, naturally, possible.
The DM could, of course, houserule that a repeatedly-summoned creature soaks up experience like a cohort or additional party member would, but that's in houserule territory beyond the scope of this variant rule. The loss of experience would be more detrimental than for a typical cohort as cohorts are able to always be present, putting their soaked exp to constant use, while summoned creatures are limited by spell slots. In exchange, you would have disproportionately powerful summon spells, as e.g. Summon Monster II could eventually summon a Celestial Eagle Fighter 5 instead of the typical CR 1/2 Celestial Eagle.
It's worth noting that the variant rule says "it entails some special problems, so don't allow [this] without considering it carefully." Allowing progression on top of the variant rule exacerbates most of the problems, and I personally wouldn't allow it barring a very specific situation.
