The canon reasons for throwing the SLDF at the Rim Worlds itself are generally pretty sound.
The reasons for taking the Hegemony world-by-world are not.
Let's bring up the WWII comparison. The key point of the Island Hopping campaign in the Pacific was that bases heavily fortified by the Japanese were isolated and ignored. The key Japanese bases of Truk and Rabaul were still in Japanese hands when they surrendered, they'd simply been rendered insignificant.
The "needs supply bases" issue is often raised, but the realities of BattleTech's technology renders it moot. So long as the SLDF could take territory anywhere on Terra, it would have its supply base. And without the campaign to tackle the entire Hegemony, the SLDF, and particularly the SL Navy, would have been in much better shape. Meanwhile, Amaris would have had years less time to prepare his defenses -- both the SDS and the conscript troops. The SLDF, meanwhile, would have its full force available.
Once the SLN seizes the jump points in the Terra System, and clears the way to the planets, the rest of the SLN could concentrate at the jump points, both the standard ones and the predictable "pirate points" (and make no mistake; they're utterly predictable; it's called celestial mechanics, and no system's celestial mechanics would be better understood than that of Sol). Naturally, a sizeable reserve would be maintained in case something odd happened. But the SLN was still large enough that it could take the entire Rim Worlds Navy (or Amaris Imperial Navy, if you prefer) at one time and not bat an eye. It's not like the Usurper's troops can easily bring their SDSes from other worlds, and those seemingly did the worst damage to the SLN during the years-long campaign to retake the Hegemony world-by-world. So the SLDF would have an overwhelming naval superiority at Terra, had Kerensky attacked at once.
The SLDF was also still a finely trained instrument, and Amaris's forces were less so. By giving Amaris nearly a decade to train and prepare for the obviously-inevitable Battle of Terra, Kerensky gave Amaris plenty of time to hire additional mercenaries, make use of the Hegemony's highly-advanced technology to reequip his forces, and create large conscript armies. Kerensky played directly to Amaris's advantage, sapping the SLDF of irreplaceable men & equipment while giving Amaris plenty of time to keep manufacturing and training his own units.
Now, the other favorite concept, psychology. Read Sunzi's Art of War some time; in it, the ancient Chinese general advises to always leave an enemy a way to retreat. Sounds odd, at first, but it has a sound reason for it: people who feel trapped rightly recognize that they have two options: surrender or fight as if their lives depend on it. The SLDF doesn't appear to have taken a whole lot of surrenders, and by cornering Amaris and his troops he gives them more motivation to fight hard, without breaking their will. So by striking Terra immediately after leaving the RWR, Kerensky achieves strategic surprise, with all its well-known mentally-unsettling effects ("Wait, doesn't he have to take the other worlds, first? What's he doing here!"), while his Navy can assure that Amaris will not receive any reinforcements from the RWR troops uselessly garrisoning the rest of the Hegemony.
The fight for Terra would still be hard, but the SLDF would have been in much better shape to deal with it, there would have been many, many fewer losses to the SDS system. Amaris would doubtless have tried to hold the rest of the Hegemony hostage, so the first order of business for the SLN would have been to conduct orbital strikes on every HPG in the Sol system, to prevent Amaris from issuing further orders to the rest of the Hegemony. This might have taken long enough for him to still send out genocide orders, but there's no guarantee his troops would have followed such orders, anyway, and once you start genociding people, they tend to fight back -- for that same "you're going to kill me if I stand here, so I might as well die trying to kill you before you kill me" result. Which means you could well have popular uprisings, and not a few Amaris units rebelling and killing their commissars (there are only a few commissars, after all, and a lot of soldiers). Given that H:LoT I indicated there was a lot of shock and outrage at the Usurption among Amaris's own troops (who had nevertheless just seized the Hegemony), the chance for significant defections from the RWR/AI forces is very large.
So, checkmate in two moves.
Once Amaris was dead -- and thus the proverbial snake decapitated -- how many of the remaining RWR troops would have remained loyal to him? Likely not very many. And Kerensky would have plenty of troops still alive to handle the few hardliners, while many, possibly most, of the Hegemony worlds would have been retaken without a fight. The SLDF would have been quite strong enough then to awe the Council Lords, and potentially return to the Periphery to force them back into line -- they certainly didn't have any significant numbers of troops remaining, anyway, per the canon numbers in H:LoT I.
Would this all have worked? I don't know, but I think it would have resulted in less devastation than Kerensky's canonical campaign. Kerensky approached the canon campaign in a very direct, unimaginative manner. "I'm going to take every territory in the Hegemony until I slowly get to Terra." Rather than strike the snake's head and let the body wither, Kerensky started at the tail and worked his way up to the head.
Kerensky was evidently an avid player of Risk, and not of chess.
