This would require extra work.. But dnd 5e has backgrounds to choose. A DM can use your backgrounds for roleplayer/questing.
Just make some quests/dialogue be based on your specific background. Eg if your background is criminal you must choose a particular criminal organisation/in a certain town. When you go to that town you can use your contacts for meaningful dialogue/role-playing options.
The same process can be done for most (or all) backgrounds. Add alignment and personality to other party members there will be no robotic companions.
So if you are a chaotic evil rogue and all the available companions are lawful good, that's a problem. They shouldn't stay in your for too long.
More options, more fun