There's a huge issue with your premise... you're asking for "a completely unbiased xp system" - by its very nature, an XP system will be biased towards "actions which result in earning XP" and against "all other actions." Even if we take as an extremely simple measure the traditional three pillars of D&D (combat, social interaction, exploration), there's no system that works well for all of them.
- Old-school D&D with its 1 XP per 1 GP model is biased toward "treasure collection" (stealing treasure/fooling guardians is basically as good as killing them) - there are no direct rewards for combat, social interaction, nor exploration per se.
- Third edition introduced the XP-versus-CR system, which is biased towards "overcoming obstacles" and in theory could do a better job but with weak social interaction and exploration CR rules, tended to be biased heavily in favor of combat.
- Fifth edition suffers from many of the same issues as third.
As mentioned early in the thread, the Palladium XP system does actually do a pretty good job of coming up with what I would consider close to an unbiased system (as it still has subjectivity on the part of the GM but tries to explicitly reward as many actions as possible).
For Class and Level, I find the Palladium list surprisingly good:
It is, however, full of subjective elements. What constitutes a minor/major/great menace is in fact supposed to be relative. I've used it in Palladium Fantasy and Robotech in the past.
25 Performing a skill (successful or not)
25 Clever, but futile idea.
100 Clever, useful idea or action.
100 Quick thinking idea or action.
200 A critical plan or action that saves the character's life and/or a few comrades.
400-1000 Critical plan or action that saves the entire group or many people.
100-300 Endangering the character's own life to help others.
500-700 Self-sacrifice (or potential self-sacrifice) in a life and death situation (like leaping in front of a fireball meant for someone else to save that person, even Though he/she is likely to die, or offering his/her own life to save the group or another)
50-100 Avoiding unnecessary violence.
100-200 Deductive reasoning and/or insight.
50 Good judgment.
50 Playing in character bonus.
50-100 Daring or heroic (whether clever or not).
25-50 Killing or subduing a minor menace.
75- 100 Killing or subduing a major menace.
150-300 Killing or subduing a great menace.
... but even this system is still biased since a "Critical Plan or action" is rewarded more than "Avoiding unnecessary violence" or "performing a skill" - the actions that get bigger rewards are those towards which the system is biased (you might agree that this is a desirable bias, but it is biased nonetheless).
The system I think is the "least biased" is probably the "participation award" model - every campaign session your character takes part in and survives gives you, say 200 XP - this is biased toward "in-campaign actions" but is agnostic as to what kind of actions it is rewarding, and I do think that rewarding players that attend sessions more XP than players that don't is a desirable bias. But it's still biased!
I think perhaps the "fairest" system is the goals/rewards model - each PC should, before the session starts, be furnished with several goals and associated rewards. If the player accomplishes a given goal during a session, they receive the associated reward (usually, but not always, XP) at the end of the session. This gives the players agency as to which goals they want to pursue and how they choose to prioritize those goals, and the only subjectivity from the GM is based in setting up the initial reward - once you've given the players "if you do X, you will get Y" you allow them to weigh the relative value of the rewards and prioritize their actions accordingly. This could easily be mixed with other models (for instance, you could have a standing set of "general" rewards such as "survive the session / get 200 XP" or "play in character / get 50 XP" or what have you) but I think this method gives the players the most agency around their advancement and lets them decide what to prioritize in order to drive the story forward, rather than having the DM decide priorities and hand out XP at the end of the session subjectively based on the PCs' actions - instead, the DM decides to weight rewards ahead of time and the players objectively select goals to pursue with that knowledge in mind. Bonus points when players' goals are mutually exclusive, so if Player 1 accomplishes Goal X, player 2 cannot accomplish Goal Y (this is best used in moderation so that parties have to make compromises and realize they cannot do everything, rather than using it to set up conflicts - for instance, I would suggest against goals like "kill NPC Bob" and "prevent NPC Bob from dying" and instead use goals like "deliver this MacGuffin to the village 2 days' travel East from here within 2 days" and "deliver this message to the village 2 days' travel West from here within 2 days" - the players will have to decide which one takes priority - or perhaps they choose to "rescue this NPC in the village 2 days' North from here within 3 days") - in this case, you are driving "meaningful choices" rather than "antagonistic intra-party relationships" and PCs might prioritize the MacGuffin this week for Player A's reward with an understanding that next time there is such a fork in the road, they will prioritize Player B's or Player C's reward.
Hopefully that made sense on the page; it made sense in my head.
