Star Control 2 had a "race against time" plot that actually had implications within the mechanics of the game. If you took too long to win, the Ur Quan Kohr-Ah would begin their death march across the galaxy, wiping out everyone species by species. Once they got to earth, you lost. This would probably be harder to to do in the context of ME1, though. Because the Reapers are so badass that their entrance into the galaxy signals guaranteed doom for all, you can't have that gradual progression towards the failure state: If Sovereign hasn't activated the Citadel relay yet, you can still win, but once he does, you lose. There's no way to implement the race against time mechanically except as a global timer that, once it expires, signals automatic critical mission failure. Not particularly satisfying.
As far as space exploration goes, I agree that it wasn't that successful in ME1, mostly because it wasn't essential as a game mechanic. Exploration within hubs like Noveria and the Citadel, however, worked much better than it did in later installments IMO. The size of these places added some much-needed verisimilitude to the proceedings, and you didn't have every combat zone flagged by the presence of chest-high cover. I think later installments could have benefited from having a smaller number of more fully realized locations, rather than having a large number of missions on one-off locations like Aite, Zorya, Haestrom, Palaven, etc.