First let's ask whether I'm wrong for assuming that the cut off frequency doesn't play any role if the overall gain bandwidth is sufficient enough? In other words, we can increase the gain to more than what we want, in order to lower the cut off frequency, and still get the same unity gain bandwidth, and I think the opamp should behave the same since UGBW is constant, shouldn't it?
Regarding the offset, since I tested it with an ideal opamp, I think offset is not an issue.