How to reach $\aleph_1$ without power sets?

At the end of Set Theory and the Continuum Hypothesis, Paul Cohen wrote:

A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the Axiom of Infinity is probably that we feel it absurd to think that the process of adding only one set a a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now $\aleph_1$ is the set of countable ordinals and this is merely a special and the simplest way of generating a higher cardinal. The set $\mathfrak c$ is, in contrast, generated by a totally new and more powerful principle, namely the Power Set Axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up to that cardinal from ideas deriving from the Replacement Axiom can ever reach $\mathfrak c$. [...]

This seems to imply that Cohen thought that building up to $\aleph_1$ "from ideas deriving from the Replacement Axiom" can reasonably be expected to succeed. But how?

It is definitely not the case that one can prove that $\omega_1$ exists in the theory we get from naively omitting the Power Set Axiom from a standard presentation of ZFC -- for the set of hereditarily countable sets is a model of that theory and contains no set of all countable ordinals.

Cohen was certainly well aware of this, so he must have had something other than that in mind. Short of resurrecting him so we can ask him, is there any evidence that shows which kind of "building-up" he was thinking of here? Is the argument simply that if Power Set had not been invented, we should still have accepted "$\omega_1$ exists" as a new axiom for a reason similar to the one he proposes for accepting Infinity?

I always took that paragraph to mean the following: In the theory (ZFC - Power set) + (Every cardinal has a successor), we cannot prove the existence of the set of reals (for example, this holds in $H_{\kappa}$ for a weakly inaccessible $\kappa \leq \mathfrak{c}$). In this sense, power set axiom is a more powerful principle than "every cardinal has a successor". Foreman and Woodin gave a model where this holds globally: For every $\kappa$, $2^{\kappa}$ is weakly inaccessible.