I believe you're overthinking what Chow means by "almost provable." We have an arithmetic statement of the form $$(*)\quad\forall x\varphi(x),$$ which while not provable in PA has the property that for each individual natural number $n$ the instance $$(*)_n\quad\varphi(\underline{n})$$ is provable in PA (where "$\underline{n}$" is the numeral corresponding to the natural number $n$). I believe this is what "almost provable" means in this context; perhaps "locally provable" or "instancewise provable" would be better terms, but I think that's all there is to this.

Note that we also have the same phenomenon with respect to consistency: for each natural number $n$, PA proves "there is no PA-proof of "$0=1$" of length $<n$." So PA almost proves Con(PA).

You also ask some questions about infinitary content and interpreting ordinals. Here I don't really understand what you're getting at, unfortunately (and this might be the cause of the downvote - I find the entire paragraph beginning "Interesting so far" to be quite confusing, and I'm not at all sure that there's a real mathematical question here).

That said, let me make a couple comments:

• When you ask "given Prof. Chow's "list" notation of "ordinals below $\epsilon_0$", how can that be extended to include $\epsilon_0$ as a "special type of finite list of finite lists of finite lists of … of finite lists,"" I think you're misreading Chow - earlier in that answer (in the paragraph beginning "finally") he is careful to say that by "ordinal" he means "ordinal smaller than $\epsilon_0$." So his statement that ordinals can be represented as [stuff] isn't meant to apply to $\epsilon_0$, since the term "ordinal" is being used there in a restrictive context.

• As to "infinitary content," I think this part of the question boils down to a confusion around what that means - different authors use the term in different ways. The simplest way to approach this is to identify finitary mathematics with some specific theory - say, PA - and then by fiat everything not provable in this theory is infinitary. You don't seem to adopt this interpretation, per your comment "it must, in some sense, be 'infinitary', that is, its proof must somehow "assume the existence of infinite sets."" Frankly I think this is reasonable, and it seems silly to me to identify finitism with a particular formal theory, but the point is that people can disagree over whether a given piece mathematics is finitary. With this in mind I think this aspect of your question isn't really mathematical, and you'll have to settle for a soft answer - namely, that in some sense when we add Theorem 2 to PA as a new axiom we "get new ordinals." (And of course this is hiding an underlying assumption - that regardless of whether we identify finitistic mathematics with a specific formal theory, we are fixing a notion of "finitistically justifiable" ordinals, namely those $<\epsilon_0$. If one holds that all ordinals $<\Gamma_0$ are "finitistically justifiable" then the above breaks down.)

I believe you're overthinking what Chow means by "almost provable." We have an arithmetic statement of the form $$(*)\quad\forall x\varphi(x),$$ which while not provable in PA has the property that for each individual natural number $n$ the instance $$(*)_n\quad\varphi(\underline{n})$$ is provable in PA (where "$\underline{n}$" is the numeral corresponding to the natural number $n$). I believe this is what "almost provable" means in this context; perhaps "locally provable" or "instancewise provable" would be better terms, but I think that's all there is to this.

Note that we also have the same phenomenon with respect to consistency: for each natural number $n$, PA proves "there is no PA-proof of "$0=1$" of length $<n$." So PA almost proves Con(PA).

You also ask some questions about infinitary content and interpreting ordinals. Here I don't really understand what you're getting at, unfortunately (and this might be the cause of the downvote - I find the entire paragraph beginning "Interesting so far" to be quite confusing, and I'm not at all sure that there's a real mathematical question here).

That said, let me make a couple comments:

• When you ask "given Prof. Chow's "list" notation of "ordinals below $\epsilon_0$", how can that be extended to include $\epsilon_0$ as a "special type of finite list of finite lists of finite lists of … of finite lists,"" I think you're misreading Chow - earlier in that answer (in the paragraph beginning "finally") he is careful to say that by "ordinal" he means "ordinal smaller than $\epsilon_0$." So his statement that ordinals can be represented as [stuff] isn't meant to apply to $\epsilon_0$, since the term "ordinal" is being used there in a restrictive context.

• As to "infinitary content," I think this part of the question boils down to a confusion around what that means - different authors use the term in different ways. The simplest way to approach this is to identify finitary mathematics with some specific theory - say, PA - and then by fiat everything not provable in this theory is infinitary. You don't seem to adopt this interpretation, per your comment "it must, in some sense, be 'infinitary', that is, its proof must somehow "assume the existence of infinite sets."" Frankly I think this is reasonable, and it seems silly to me to identify finitism with a particular formal theory, but the point is that people can disagree over whether a given piece mathematics is finitary. With this in mind I think this aspect of your question isn't really mathematical, and you'll have to settle for a soft answer - namely, that in some sense when we add Theorem 2 to PA as a new axiom we "get new ordinals." (And of course this is hiding an underlying assumption - that regardless of whether we identify finitistic mathematics with a specific formal theory, we are fixing a notion of "finitistically justifiable" ordinals, namely those $<\epsilon_0$. If one holds that all ordinals $<\Gamma_0$ are "finitistically justifiable" then the above breaks down.)