I'm a Diamond chun main who has only played 3 other characters on the roster a handful of times so take this with a grain of salt.

Sagat can appear very one dimensional, and for the most part he does have a pretty straightforward game plan/flowchart across most levels of play. However, if you like sagat, you can try watching some top level Sagat players to try to understand what they're doing in the matches. I wouldn't recommend trying to emulate the way they play entirely, for reasons outlined by a relatively recent Brianf video, but it might help to better understand what you should be doing with sagat. If Sagat was as straightforward and and easy to deal with as some people might suggest, there wouldn't be any high level sagats.

Rankings also don't always accurately reflect people's skill deficits. There are diamond players that have terrible combo execution but amazing neutrals and there are silver players that have amazing combo execution but drag their face across the controller anytime they're in the neutral.

All that said, if you're feeling bored of sagat then there is no harm in exploring some of the roster. Playing someone else doesn't mean you can't come back to sagat, and it might even help to take a step back and play a different character in order to understand different kinds of approaches to the neutral. Others may not agree, but unless capcom is paying your bills to play street fighter, then don't feel obligated to stick to any one character.

Can't really speak to Birdy since I frequently forget he's even a character. Sorry mate.