The catch I'd expect here is for the marginal immunological benefit from an extra cold to be less than the marginal cost of suffering an extra cold, although a priori I'm not sure which way a cost-benefit analysis would go.

It'd depend on how well colds help your immune system fight other diseases; the expected marginal number of colds prevented per extra cold suffered; the risk of longer-term side effects of colds; how the cost of getting sick changes with age (which you mentioned); the chance that you'll mistakenly catch something else (like influenza) if you try to catch someone else's cold; and the doloric cost of suffering through a cold. One might have to trawl through epidemiology papers to put usable numbers on these.

Consuming probiotics (or even specks of dirt picked up from the ground) might be easier & safer.