An all party approach seems sensible, even if it is motivated by cynical aims (ie so it doesn't get seen as a 'Conservative' problem).
I can see social care being a major issue in the 2024 election, and it's one which could hit the Conservatives older voting base. A lot of the 55 to 65 year old generation are now dealing with elderly parents in social care and seeing just how difficult the current situation is. Social care costs will also rise if the loss of migrant workers drives up wages, which will be welcome in a low-wage sector, but someone has to foot the bill.
Either this means that individuals have to meet the costs of their own care (eg from selling their homes) or caring responsibilities should fall to other members of the family, or it needs to be paid for out of higher taxation. Given the huge (unearned) windfalls that people have earned through house price appreciation it's not unreasonable to start there, but that won't be popular with Conservative voters, neither will they like the idea that they are responsible for caring for elderly relatives (why not? other cultures do it?). And of course there will be lots of right wing lobby groups that will pile heat on to the Chancellor if he is tempted to raise taxes.
So it's quite likely that the can will be kicked down the road with the system getting steadily worse until it becomes a real political weakness for the government. If the Conservatives were thinking that there was a chance Labour might come in in 5 years time the best tactic might be to just mark time and let Labour take the unpopular decisions required to sort it out, and then attack them for it, but as they are probably thinking of a 10 year stint in power if not 15, they will have to address it themselves at some point.
