^Praetorium is useless for farming soldiery. With a good team the fastest way other than finding new people to do dungeons with is to farm snowcloak which is 50 a pop. But even then people don't really bother with it tbh. I do enough of them running my roulettes during the week and I cba to farm it.
I clear about 800 soldiery a day. You're guaranteed about 500 from your daily roulettes and if you get lucky with new players (which I don't) you can hit almost 1000 just from the roulettes, so it becomes super easy to gear out your class in 100 gear by the end of a week.
Then farm Syrcus (which also gives soldiery) for sands and oils of time to upgrade the 100 gear to 110. It only took me about 5-7 days to do this.
The expert dungeons now drop level 90 gear and "vintage head/body/breeches/arms" etc that you can swap for the relic gear which is also level 90, I've now got a set of that gear sitting waiting for my WHM and WAR.
Syrcus Tower also drops level 100 gear.
The main problem with this stuff isn't the time investment - because by MMO standards it honestly doesn't take long at all - it's that people haven't completed their main story and sidequests to unlock the roulette. You need to do the Hildibrand stuff aswell because that unlocks Gilgamesh and Ultros which are required for the trials roulette, and the trials roulette is my favourite because the fights are all fun and like you said it takes like 5 minutes.
If you do this you'll definitely hit the weekly poetics cap aswell which allows you to buy the 120 gear. Carbontwine obtained from Final Coil upgrades this to 130 gear.
Weapons are another issue unfortunately. The relic weapon is always going to give you the best stats when fully upgraded (even better than the once a week coil drops) and it's not difficult but it is definitely the most time consuming thing I've done and I literally have weapons that are the same iLvls for my other classes already.
Steps are (after the long quest to actually get the relic weapon):
Zenith: Buy 3 Thavnarian thingies from revenants toll to upgrade the weapon at Gerolt's bit in north shroud
Atma: Farm atma in 12 areas from fates. Atma of the Scorpion drops in Southern Than etc. Drop rate was 2.5% in patch 2.3, is now 5%. This one really is just RNG and luck. I got two after one or two fates in certain areas, and in Central Thanalan I was literally doing every fate that popped for two whole days. Central Shroud took about 6 hours etc. It's tough.
Animus: 9 Books which cost 500 Soldiery (used to be Myth) each that need to be completed. One book can be done in one session of gaming aside from the fates. Each book has 10 enemies, 10 of which each need to be killed, so that's 100. Then you need 3 fates. The fates are often rare or on a 2-3 hour respawn timer. Then you need to complete 3 levequests, and finally 3 dungeons. You need to defeat the boss in each dungeon, and if you die and the party defeats the boss, this doesn't count so beware of this.
Novus: You are given a sphere scroll with 75 slots and you need to use alexandrite with a materia to upgrade it. The melds can fail, but if they fail you don't lose the alexandrite. You do lose the materia. The neat thing about this is that you can choose your own stats. Alexandrite is of course, ridiculously hard to come by. There's a daily quest in Mor Dhona that requires you to do the expert roulette (which you should be doing anyway) and gives you a mysterious map. Mysterious maps require you to find a location and dig, and digging nets you a battle. When you win the battle you get a chest, and the chest contains the alexandrite. Mysterious maps are also sold by auriana for 400 soldiery, so that's where my soldiery goes now because this is the stage I'm at. I usually get about 3 or 4 alexandrite a day. Also of concern is the materia, which costs a lot. You're going for specific materia, so for me it's savage might materia etc. The amount you need of each class varies depending on the stat. So if I want to max crit hit rate (I do), I need 11 of Savage Might 1, 2, 3, and 4. I, II and even III are affordable. Savage Might IV sells for something like 600,000 Gil minimum on the market board and the only other way to get level IV materia is to convert fully spirit bound weapons and armour that are level 60+ and hope you get lucky. Another way to do it is to just use up to level III materia and put some points into a tertiary attribute (for me that would be accuracy), so this is probably what I'm going to do.
Nexus: You need to reach 2000 "light" from defeating the trials such as primals. This seems to be the least evil (to me) of the bunch. Ifrit hard gives about 16 light for example, and some have bonuses. There's a tracker available online. I'm looking forward to this step.
The next stage is going to be in 2.5.
So yeah, 100% will not be doing all that for an alt class.