Why would it be forbidden?
If the prohibition stems from the principle of afflicting the soul, and the reason is that leather shoes are supposed to be more comfortable,
we know that today most shoes, including those not made of leather, are designed to be comfortable.
So a comfortable non-leather shoe is permitted, but an uncomfortable leather shoe is forbidden?
I’m asking because I actually do have a relatively uncomfortable leather shoe.

Although there is no Talmudic source that the prohibition is specifically on leather shoes, this argument only means that there is a prohibition on all shoes, not only leather. It does not permit leather. The fact that you have an uncomfortable leather shoe is a private case, and it is not reasonable to establish a unique halakhic ruling just for you. As a rule, one does not walk around in regular, standard shoes. Therefore I really do think it is not correct to wear rubber shoes that people normally wear in everyday life.
Beyond that, if your shoe is uncomfortable, why did you buy it? Apparently it is comfortable for you.

Is it permitted to wear flip-flops, from the standpoint of clothing that is not fit for prayer? (I wouldn’t stand like that before an important person, but I don’t have any other footwear that is permitted.)

Yes. These are the laws of Yom Kippur.