The ideology of Hamas may live on I agree but probably not as far as the government of Gaza is concerned.
The centre of Gaza has been split by the Israeli army, it's only going to go one way. To be honest I thought this would be a nightmare urban long war spread over years but the ground campaign has so far been rapid. As it's impossible for Hamas to get arms in it's going to be effectively over with very quickly.
The alternative is otherwise as you say they back off, tolerate Hamas and remain at war forever, either suits Israel. This is because the fact is that Israel is a well supported and probably nuclear armed country whereas Hamas definitely is not any of these so there is only ever one winner.
Perpetual ideological war is perhaps inevitable but the outcome of this physical war is also inevitable. It is very unlikely that Hamas will exist in its current form within 3-4 weeks time. They might reset and rebuild but I don't see how the Palestinians benefit from this cycle so it only makes sense to remove Hamas and keep them removed.
I'm arguing to move the Palestinians temporarily, that is not ethnic cleansing. If Israel invades Gaza and claims it for its own then I will agree, that hasn't happened.
That is unless you think it's better, safer and more appropriate for the Palestinians to live amongst people who decide to build their military infrastructure beneath hospitals?