I've done plenty of monoclass runs to be honest.
Barbarian, Fighter, Druid, Cleric, Warlock, Sorcerer, Wizard and Bard are all able to be very strong as monoclass builds. Paladin, Monk, Rogue, Ranger can work but tend to benefit from multiclassing.
There's plenty of options there for parties. Even if you try to stick with companions having their original classes, you've got:
Shadowheart as a Cleric (Light Domain or Tempest Domain are both strong)
Lae'zel as Fighter (Battlemaster is solid, Eldritch Knight Tosser build is very strong)
Karlach as Barbarian (Tiger + Tiger Wildheart is good. Berserker Tosser build is decent)
Gale as Wizard (Evoker, Diviner, Abjurer, Necromancer are all strong)
Wyll as Warlock (Pact of the Chain and Pact of the Blade are both viable)
Astarian suffers as a Rogue, but you can work with Thief or Arcane Trickster.
Then if you're not wanting to double up on classes, you've options for Tav;
Moon, Land and Spores Druid are good.
Draconic and Storm Sorcerer is good.
Lore and Swords Bard are great.
The only real thing you have to think about, is making sure to have access to Longstrider (And to a lesser extent, Leap) as you can't get this from an item like you can Feather Fall (From the boots in the Arcane Tower) and Guidance (From the Harper amulet near the Grove) and is very nice to have (Yes, there's the Ring of Jumping, but it's self cast only and once per long rest)
