I don't think that anything should be means tested. Taxation of income (particularly unearned income) is much fairer, and would, if properly and rigorously applied, bring in enough to pay for both nurseries and care in later life.
Means testing penalises thrift, and removes the freedom to do as you wish with money you have earned. It also works against social mobility, as those who have just above whatever threshold is applied are not able to pass on money to the next generation to give them a better start in the way that the better off can. Each generation has to start from scratch. Taxation of inheritance would even that out, and be a lot fairer across the board.
Means testing puts all of the power in a household into the hands of the higher earner, as the whole household is denied means tested 'things' which the earner may be able to afford to buy, but not allow others to do. In functioning households this won't a problem, but it can be very difficult for members of dysfunctional ones to manage.
Personal pensions are not free, and if anyone who has saved into one is denied the same care/services/benefits as those who don't, then what is the point of saving? Again, means-testing in this way penalises those who do without in the short term in order to provide for later.
So the short answer, IMO, is yes - we should fund care at both ends of life, paid for from proper taxation at source, and not penalise those who, after tax, have saved for whatever they choose in their old age, whether that is care, world cruises, gifts to donkey sanctuaries or money for their children.