From what I understand, a balanced D&D party needs for archtypes: A melee fighter for dealing with other melee fighters, a rogue for detecting and removing traps and locks, a caster for dealing with enemy casters, and a healer to keep everyone standing.
A party limit of four really completely kills gameplay diversity in my mind. Are there any good multiclasses you can do with a level limit of 10 that can let a character handle two of those roles without crippling themselves at one of them?
That is a very strict approach, and definitely doesn't apply to a tabletop environment. A more structured video game obviously allows for less creative flexing in your party, but you still don't need to be quite as rigid with your party roles. In particular, having a rogue be its own archetype is overkill - you don't HAVE to sneak around everything and disable every trap, and a party can succeed just fine without it. You also don't need a rogue to fill that role. Rangers, Druids, Bards, Monks, Clerics, and Wizards all can make excellent scouts.
A dedicated healer role is usually going to be ineffective. Healing is usually far too inefficient to go all in on it, though its obviously nice to have someone in your party that can provide a bit of healing in a pinch.
Instead of traditional archetypes like Tank, Healer, or Ranged DPS, imagine each character each taking on multiple smaller jobs, often with redundancy between each other. Your Paladin can be a frontliner who also provides occasional healing and buffs to your team. Your Ranger could be another frontliner with more mobilitiy that also makes a good scout, and can swap to ranged combat when needed. Your Wizard can be a long ranged blaster that also can detect traps with his high investigation, as well as providing lots of utility and battlefield control. Your Cleric can provide healing for the party as well as consistent midrange damage, while also providing invisibility and stealth bonuses to sneak past guards.
You can be very successful with a huge variety of party compositions, and this is before you even attempt to start multiclassing.
