I've seen GMs use random tables. Have you ever seen a player use random tables to play? I was play testing a game I'm writing, using my regular players. My wife decided her character's powers all came from the Chaos Goddess Catastrophe.

She then proceeded to decide each time to make decisions about which way to go using random dice rolls. Her sister and our other friend both decided to let her do that. Except once the other friend, who does a really good Russian accent, said "That is silly. We will go in this door" and she opened the door they were in front of. (Picture a lichen katydid carrying a two handed axe and a crossbow, and speaking with a Russian accent.)

My sister-in-law played most of the game in dog form (her character is a werewolf). Fortunately, she has a telepathy spell and the katydid has a speak to animals spell. She was faster than the other characters and would run to a door and check it for traps and whatnot, to try to short circuit the Chaos follower's die rolls.

Things I learned from the playtest:

I accidentally dropped the healing rules from the rules during a rewrite.

I need places on the character sheet to keep track of the number of spells cast.

I forgot to include a default morale table for monsters. (Percent chance to run based on death of companions.)

I forgot rules/guidelines for stuck doors and destroying parts of rooms.

I need more setting specific "choose from this list" for some parts of character creation. My sister-in-law was overwhelmed with a couple of open-ended sections. I want to do a Traveller-style life path and/or maybe a character-concept generator, to give more help.

Yeah, I have and have had players who roll dice for their own character's decisions that they could choose as players. I have too, especially when there isn't a lot of detailed GM descrition, or it's an incidental detail or something I just don't feel I have access to what the PC is experiencing or would do or prefer. Particularly for characters with (GURPS) Disadvantages I don't think I have, such as behavior imbalances that don't always drive behavior (e.g. addictions, paranoia, short attention span) or senses (e.g. the GM is describing what the group can perceive, but especially if my character is unobservant, distracted, or hard of hearing or short-sighted or something, then I make my own roll to see if my PC notices), and reaction rolls for when I have no clear sense what my PC would think of some people, or whom they'd be attracted to (or just to practice the reaction roll rules which I usually don't have time to think about all the DRMs for as GM), and also just for general arbitrary whims and choices.

Skarg, I built a monster reactions table for sapients, but their random encounters and room encounters haven't brought them to any sapient beings. She was using this method because the trail they were following disappeared as they entered a dungeon. At that point, there wasn't a really good choice of direction so...