If omicron does not overwhelm hospitals, will it be allowed to spread?

The implication of "allowed" is that they have the choice or ability to do otherwise. The most effective tool against covid-19, the vaccine, has already been widely deployed. Most other solutions are generally just virtue and in-group signaling.

Politicians are followers, not leaders, they do what they think their constituents want. Ultimately what action they take depends where you are.

Omicron will overwhelm hospitals.

I'm not sure what you mean by "allowed to spread" -- people didn't want to lock down for alpha or delta, and they're no appetite for lockdown now, especially for something like omicron which is widely (but mistakenly) perceived to be "less dangerous" than delta. Because of the increased transmissibility any lockdown would have to be pretty fierce.

If omicron is really so mild that it isn't putting anyone into hospitals and the spread of omicron also eliminates Delta, then yeah governments will probably stop bothering with measures to contain covid.

These are really, really big assumptions though.

Allowed? It will spread regardless of what we do.

It will not significantly overwhelm hospital either.