I cannot even imagine gaining more than one level in a month.
Yes, it seems far too overheated, yet that is what 5e RAW entails once you work out how many rests would be needed. A key factor in protracting that would be travel time; so the campaign type plays a big part. For a party in a multi-level dungeon, travel time might be low. For those in an open-campaign, it could be months or years. I favour the latter.
So I was just drawing attention to what the RAW makes possible: not my preference.
Yes, it is impossible to predict who will become a hero, but the more powerful that person is, the more predictable the hero's actions are. So you can plan a strategy based on what types of actions heroes of different types can perform, and prepare counteractions in advance.
I see. As to that particular question, I can only speak from my experience DMing and playing. I have
never experienced players becoming
more predictable as they gain power: universally the opposite. One reason might be that as they gain leverage over the narrative - spells to acquire information, travel quickly, and dominate their environment - efforts to challenge them tend to increase, rather than decrease, the chaos.
Perhaps the safest thing would be to let them alone, which of course isn't too suitable if one is a member of an existing elite that they will supplant. Hence I suppose that a campaign could stabilise around the idea of elite organisations, that might prefer to recruit than confront such heroes.
Or at least, why would such organisations
not seem reasonable? They exist in official sources, and I effectively am suggesting that they can be used in a way that feels plausible and consistent. If not that way, what other way might be consistent for them?