This is probably the issue, actually.

OP: your fare doesn't automatically cover an F seat in this scenario, but rather a seat specifically in P class. If there was no P availability, the agents were correct to deny your request, although you could have waitlisted for P (if the agent knew how to do that).

It may make more sense if you consider that the P fare that you purchased required availability in P on each segment*. If you had tried to search for IAH-BOM and P hadn't been available on the domestic connector segment, it would have pushed the fare up to the next available fare class, potentially adding thousands of dollars to the price. So, while from your perspective, you're likely thinking "if there'd been an F cabin, it would have booked me into it automatically," it's also true that if there had been an F cabin, your flight might have been dramatically more expensive.

All that said, United is nothing if not inconsistent, and I'm not the least bit surprised that an agent was willing to override this to do the customer-friendly thing. I just think that you should consider it a favor rather than an entitlement.

*UA actually marries inventory together, so that you can't necessarily extrapolate the inventory for the IAH-DCA part of IAH-DCA-EWR-BOM from the search you get for IAH-DCA by itself, but the general point stands.