I can somewhat understand what you mean, but figuring out the badnik movements and possible base location of Eggman seems like it would make people more confident in us? Like, "Hey guys, we know where the problem is and what they're doing. We know where to focus our efforts." Feels like it communicates that we aren't incompetent. Not knowing what's going on would make us look incompetent, knowing what's going on makes it look like we're on top of things (which we hopefully will be).
Could go the other way, on the logic of "if you know what the problem is why haven't you fixed it yet?"
Really, though, I feel that helping Team Sonic puts us in a better position to deal with things than just finding
one enemy base (which isn't even guaranteed to be the main one, or permanent, or any of that stuff).
...which brings up another relevant issue: Eggman frequently moves his bases (admittedly part of this might be Sonic's habit of destroying them, but still). Finding his current base is by no means a guarantee that it will continue to be his base for more than, like, 3 turns.
And a final point that relies on narrativity: It's way too early in the story for us to find his main base; it probably isn't even on the planet. Hell, it might even be too soon for us to find a particularly relevant
local base. The most I'd expect to find without massive amounts of build-up time would be a carrier or two.
...bringing this back around to "finding Sonic's friends might be more beneficial" in a way relevant to this particular point you're pushing: What if finding Sonic's friends is a prerequisite to finding those bases you want to beeline so much? For example, one of them might have decrypted his communications and could choose to work for us, or another might point us to the nearest base as thanks for helping them with something, or... well, you get the idea.