Tl;dr Can we take a weak system $A_0$ then show
$$A_0 + Con(A_0)\implies Con(A_1)), \space A_1 + Con(A_1)\implies Con(A_2)), \space A_2 + Con(A_2) \implies \dots$$
terminating in ZFC?
My understanding of Hilbert's program was that after some crises in mathematics, people wanted to be sure ZFC didn't contain a hidden contradiction, and Hilbert's idea to do this was to prove it from weaker systems. In particular, a chain of systems might be found so that $A_0$ proves $A_1$ consistent, $A_1$ proves $A_2$ consistent, etc. so that the last system, $A_k$ is ZFC.
Godel's 2nd incompleteness theorem showed this to be impossible, as ZFC cannot prove its own consistency, so no weaker system can. However, this seems to me not a substantial problem, because weaker systems like ZF can prove the consistency of ZFC assuming the additional hypothesis Con(ZF).
So in that case, it seems like (this part of) Hilbert's program is still very much possible in spirit, as one could instead take a weak system $A_0$ then show
$$A_0 + Con(A_0)\implies Con(A_1)), \space A_1 + Con(A_1)\implies Con(A_2)), \space A_2 + Con(A_2) \implies \dots$$ $$\dots \implies Con(A_k), \space A_k + Con(A_k) \implies ZFC$$
Which is still using a weaker system to show ZFC is consistent. There doesn't seem to be any reason this wouldn't be just as convincing to show ZFC consistent as the first approach.
So I would like to know whether this approach can be used to prove ZFC consistent starting from PA. If so, I would like to know if it's possible to prove it directly from PA, without intermediate systems. I would also be interested if any even weaker systems can be used.
I would also like to know if there is any philosophical reason this might be less convincing than the first approach.
Please also correct me if I have misunderstood Hilbert's program or the theorems I mentioned.