The fastest configuration I’ve found for souls per minute is (mana generator)/The Dragon Soul/(necromancy)/(necromancy) with abyssal banner on Warlord 1. Lower difficulties will go faster, but you’ll occasionally only get two casts off. Average clear time is between 70 and 90 seconds depending on the difficulty you use (but longer if you have super low surge chance, or take your time with the battle). Mana generator can be Soothsayer, Giant Spider, or Green Slime, and one of the necromancers should be a warlock to tag the last enemy standing. I personally like the Giant Spider/TDS/Warlock/Sac Priest configuration. An empowered soothsayer will probably be slightly faster if you don’t quite have the surge chance to fill up in 2-3 matches consistently, with Aziris as one of the necromancers so you can have magic link. You can also swap out the third spot for a banshee to reach the cap much faster, and though it equates to a bit less per minute, it will improve the pacing.
If you want builds more focused on clear speed that still get souls, you’ll need to have at least decent kingdom bonuses. My fastest teams are Valk/Rowanne/Warlock/Mercy, Valk/Rowanne/Siren/Siren, Valk/Bombot/Rowanne/Hero w/imperial jewel etc. Ragnagord, Banshee, Rowanne, Acolyte is one that I rarely see used. If you are after mostly souls, though, you probably want to use a three-shot dragon soul teams. While most of those are almost twice as fast, they also get a quarter to a third of the souls per battle, which will equate to about half to two thirds the souls per minute.
Some general advice on tailoring the soul farm teams to meet your needs: Usually one “phase” of the match is the wrong speed, and you should tweak accordingly. If you are too slow to charge your TDS/Valk, bring empowered mana generation, and make sure banners are focused on your soul generator. If you are capping souls and then have to do another “loop”, swap out for more damage or lower the difficulty. If you are overkilling, up the difficulty or swap out for another necro. The TDS team is actually incredibly fast all things considered, so I’m not exactly sure what “phase” is slow for you, but hopefully these suggestions helped. I suspect your team isn’t fast enough in the startup phase because you need to fill banshee to start (or a huge amount of red on TDS), then have alignment to fill the dragon soul, at which point banshee is mana blocking purple, not allowing you to get your “loop” going.
Heres the link to the other thread, by the way: