This is the huge problem with "The Paradise Syndrome." Given that they started two months in advance, it should've been easy to divert the asteroid. They didn't need to try to do it all in one big push; they could've tugged on it with a tractor beam a little bit more each day and gradually nudged it off course. The only possible rationalization is to assume it was an exceptionally massive asteroid, with too much momentum to allow its course to be changed in time. This is also the case with the moon in "Deja Q" -- it was just too big (and they had a lot less time).
The huge problem with Paradise Syndrome
(I just finished a rewatch of TOS) is a big lack of basic understanding of celestial mechanics. The whole threat of the asteroid impact doesn't stand up to basic logic once you think about it.
At the start of the episode Spock states they must "warp out of orbit within thirty minutes" in order to rendezvous with the asteroid in time. Warp out at what speed? If you are 45 minutes late can't you just warp out faster and still make the rendezvous in time? Then, later, Spock states that their delay in leaving "made it imperative that we proceed at maximum warp speed for a period which exceeds the recommended safety margin" OK, again, maximum warp speed over a period of time. Right after the log entry, Scotty states they are traveling at Warp 9.
So, either the asteroid is traveling at speeds reaching Warp 9 (which, according to Lights of Zetar
is impossible for a natural phenomenon) or this asteroid is several light years distant from the planet. If the asteroid is that far away, what's the big rush?
However, it's stated that, without Warp, it will take 59.23 days for the Enterprise and the asteroid to reach the planet. So, where did the Enterprise travel to at Warp 9? Just how close or how far away is this rock? Shouldn't 59 days still be enough time to alter the trajectory sufficiently?
Not to mention that the Enterprise, according to A Taste of Armageddon
and General Order 24, has enough firepower to lay waste to several planetary cities. Surely this would be enough firepower to break the asteroid up into smaller, manageable pieces that would burn up upon entry.