No, I actually go for Vinland first (and the last battle is really hard with this strategy - luckily it will be the only fight in that turn to make it easier), and in my current playthrough after that the Aztecs, South Sea. Doing South Sea first is probably slightly better from an optimisation point of view - after conquering Port Royal (don't use Geronimo, she is needed to do the Aztec quest), I basically waste an action on someone's character event (sadly not Himiko or Billy - those would have been more useful). Probably best to take Guam, and then recruit the Aztecs next turn but that would leave you with a fiendishly difficult battle against Cook and Magellan though. I bailed out on that one. Manpower is simply too limited.
Kamehameha is quite useless (too limited troop numbers) and drains 4 turns before you actually get her. For a few more turns you get the Knights of the Round Table, and that is obviously the far better deal. So no point in recruiting her, before you have dealt with the Taika Empire. Technically Vinland is not needed for a rushing strategy, but you really need the manpower, Alwilda is pretty awesome, and Teach has 9 speed which is very useful at times as well.
After getting Montezuma, rush through the South Seas, and take Saigon as soon as possible (Oda can be really useful for that one). Then it is time to take on the Incas, up to Valparaiso, and take on Britannia / India as soon as you can. Ideally you do the three Britannia events in the same turn, so that you can send your A-team for that battle (since the two other events do not involve fighting). Before taking on Britannia, I invaded Australia (which technically is suboptimal, since that can wait until after you conquer China). Again, manpower will really help, and Jeanne d'Arc is pretty useful to provide criticals.
The Britannia battle against Balin, Tristan, Palamedes, Percival and Mordred is extremely tough. I massively lucked out on that one. Balin beserked Columbus, and for some reason afterwards no one bothered Columbus as she blew the middle row away(Tristan and Palamedes are not dangerous - in fact you could even aim to feed the enemy brave if Palamedes / Tristan are up next, so they can waste the brave on poison mist / that luck reduction attack - sometimes it is also a good idea to feed enemies brave if they will launch a brave consuming attack with high wait time). It is Mordred and Balin in particular who are nuisances), I pretty much would have accepted any victory without units getting killed. Columbus and Benkei (Benkei's Full Swing brave attack + Columbus work great on Payens) really saved my skin in that one and the follow up. Also it helps massively to save the game during the conversation AFTER the first battle, so that you can retry the second one, without having to go through the first one. Because sometimes you just get victories that are really not great (too many units killed, too few bonuses etc). Yes, I was outnumbered, but not too badly (all fighting units were above 2000)
Notice that after you have defeated that gang, it will take them two turns to land in India. Which is a good time to declare war on Macedonia, and wrap up some loose ends (eg. conquer a few more Inca cities, or Australian ones). Then deal with India, and you'll get your second healer in Bors. 3 brave is a lot, but healing 3000 units per go is quite useful, as you will be taking lots of damage by now.
After dealing with Britannia / India, you just steamroll through Macedonia, wrap up things against the Incas.Then take on China. China is big, but it provides just a few tough battles (the ones with Lu Bu). After conquering the Taikas and ending your turn you get your fourth action point. Then you can relax a bit - you probably will need to do some character events. I really don't see how you could win the battle in Carthage with just basic attacks, let alone some of the battles in the Dark Realm. Egypt might be doable, but you are not going to get extremely useful units from there.
Another thing is you need a lot of luck not getting swamped by defensive battles. Over the course of my ten turn war with the Incas (I was actually hoping for more defensive battles, so that I could get troop bonuses for my roster, especially the ones that could use a boost) just got three against them, and no other nation, Macedonia included attacked me more than once in this playthrough. In my Japanese version run, I sometimes had to deal with 3 or 4 attacks in the same turn. Really hard if you have a limited roster - you really don't want to send out 2 or 3 units if you can help it, because they will suffer heavy casualties against fresh troops.
ETA: My first playthrough I finished on turn 73. I could not read Japanese, so I had no idea that that was indeed the final battle. Obviously reloaded that one, and did all the character events + quest before finally beating it again on turn 183 or 184.
ETA2: Also, in timeless battles ALWAYS bring a healer. Even if your units barely survive, leave one harmless unit standing, as you heal up. That might cost you a rank or two, but saving the troops is usually well worth it.