Consider the following statement:
$(*)\quad$ There are $\omega$-models of ZFC of arbitrarily large cardinality.
This is provable in ZFC + "There is an $\omega$-model of ZFC" alone. The proof I know is a "nuke:" just check that the Keisler-Morley theorem (Theorem $4.2$ here) is provable in ZFC alone.
My question is:
What's a simpler proof of this fact?
It looks to me like Keisler-Morley is doing much more work than should be necessary for this weaker result, but I don't immediately see a simpler argument.