Is it possible to combine two hash functions in such a way that cracking the constructed hash would require cracking the constituent hashes?
Suppose we try C(password) = A(password) + B(password) where + is just concatenation of the hashes.
The result of this is obviously a weaker hash: we can preimage C by finding the original password by cracking  A or B.

This is incorrect with the standard notion of preimage resistance.
Because there are many more possible inputs to a hash function than outputs, there are necessarily collisions. Even if $A$ allows you to find a preimage, that is not necessarily a preimage for $B$. For random inputs and unrelated hashes, it very likely is not.
In practical applications this is quite a bit more messy. For example, if you take $A$ to be the truncate function (to e.g. 256 bits, zero-padded), it lets you find a preimage for the concatenated hash if the input is short enough. For the vast majority of possible inputs it does not directly give you a preimage, but that is cold comfort if hashing e.g. typical passwords.

If you want a provably robust combiner, you could look into the thesis linked from an answer to the previous question: Anja Lehmann: &quot;On the security of hash function combiners&quot; (pdf). It offers good combiners that are unfortunately rather complex.
For a simpler combiner, you could use XOR like TLS does. You do not get a nice proof from only the preimage resistance of the hash, but it is provably PRF if either of the two password hashes is PRF (and they are independent). That is a practical choice that is probably good enough if you take care to use different hashes.