Eh, it has a smaller effect than you might think. If a player did absolutely nothing or was perfectly balanced in their health approach (either of which would mean their health stat should eventually approach zero) they would, on average, get sick every 18.75 days. On the other hand the literal maximum health that you can reach would be a value of 300. Just getting to said value would take you an absolute minimum of 75 days, and would require immense luck due to the way the rolls scale. At this value you would get sick on average once every 25 days.
In practice, however, once you reach 200 health than even if you are abusing every health giving thing in the game you can only reach a 50/50 chance of increasing or decreasing (and the higher you go the lower your chance of increasing and greater your chance of decreasing), meaning that the effective caps on health are at +/-200. This gives us these chances to get sick:
@-200 health: once every 14.58 days
@0 health: once every 18.75 days
@200 health: once every 22.91 days
So even if you are totally abusing health sources you only buy yourself about an extra 4 days without getting sick, or an extra week if we compare someone with totally tanked values to someone in the epitome of health.
On the other hand getting sick isn’t the only major benefit of health. Here’s the chance of spontaneously healing from a bite/infection without any treatment:
@-200 health: 38.66% chance of spontaneous healing
@0 health: 46.84% chance of spontaneous healing
@200 health: 53.76% chance of spontaneous healing
(Though if you are recovering from a previous infection it can drastically lower those values down to almost zero, depending on how close you were to death before the infection healed). Comparing the tanked values to the epitome ones you can see that you have almost a 15% greater chance of survival without treatment between the two.