You did not mention how you would distribute ability scores and what feats and proficiencies you would get, but you can't have as many as the RK above because you do not have as many feats. Also if you invested points into Wisdom and Charisma your AC is truely awful, even with shield.
Going with standard array and variant human like your Rune Knight, we can get 8 / 14 / 13+1 / 15+1 / 12 / 10. Shadow Touched gives you +1 INT, at-will access and one free cast of Invisibility and Charm Person. Take Fey Touched for the same thing with Misty Step and Cause Fear. Use the last ASI for +2 INT and you're at 20.
The reason I didn't specify any subclass is because the wizard does not need any to overshadow your Rune Knight. Be a Conjurer with a literal grab bag of options, be a Scribe and spy on enemies with your spare spellbook, be a War Wizard or Bladesinger if you want to have better defenses.
You are not nearly as good in social encounters as the Rune Knight above. It is not even close. You are closer on the exploration pillar if you have the 1st level rituals.
[cont'd]
Fly and Dimension door bring some capability for exploration, being both very limited in number of times a day and situational.
The character I put above has a +10 to stealth, a +5 on finding traps through intelligence or +8 through Wisdom or alternatively he can make a +5 perception check with advantage. He has a +10 on disarming traps and +8 or +5 with advantage on many survival checks. You don't have any of that, or really any spells to use for those things.
Moreover an 8th level Wizard who uses fly twice a day and Dimension Door twice a day out of combat is not going to be equal to a fighter in combat for 6 fights a day, especially if you invested in Wisdom and Charisma and have correspondingly very low AC.
You also should have put invisibility in there if you wanted to buff the exploration pillar. Find Familiar helps with scouting, and honestly buffs this better than either of the leveled spells you mention, but it is still not nearly as effective as someone actually good/awesome at the necessary skills.
With a 16 Charisma and proficiencies, you would be good at the social pillar, but still not nearly as good as the character I posted and you don't have a 16 if you invested into intelligence. You would be a lot better with Suggestion and Charm Person on your spell list but still short of the character I posted because the target gets a save on those spells.
Friends is highly situational, because it automatically turns the people you use it on hostile when it ends. RAW, that is specifically hostile, not just that they know you used it on them. It is also only useful against creatures not hostile, so it doesn't work for example if you are trying to convince an enemy to surrender. It is useful at times. However even when it is useful you are getting advantage using it on ONE person. You can't use friends to help you social your way past two guards because it only works on one target.
You can use Eagles splendor to get advantage, but that takes both a spell slot and your concentration. Further unless you have invested in Charisma, Charisma skill proficiencies and expertise you are still well behind the RK even when burning a spell slot and using Eagles Splendor.
The character I built gets advantage on Deception and Intimidation ALL THE TIME and has very high scores to boot. On Deception +10 and Intimidation +7 WITH advantage and he can use that advantage on groups or in situations that he can't afford to automatically turn the people he is talking to hostile or when they are enemies. You will need a 16 Charisma, and expertise in Deception AND get advantage and that is just to match the RK I posted in that one highly situational check. That RK above also has a +7 Persuasion.
Performance is the ability low one at +3, but that is why I mentioned the instrument as a tool so I could get a +9 if I had the opportunity to use an instrument.
He also has a +5 insight which is good and you have nothing for that (Detect Thoughts would work but it is not on your list).
To play a caster well means being creative. You aren't limited to rolling skill checks to make things happen outside of combat.
To distract some guards, cast Minor Illusion to mimic a screaming kid to make them leave their positions instead of trying to convince them that a kid needs help with a Deception check. To get past an area littered with traps, cast Dimension Door and walk right through instead of rolling multiple times to disarm each one.
Beyond that, I'm not sure why you place so much value in having limitless advantage on Deception and Intimidation. Do you encounter many adventuring days where you're asked to make Deception and Intimidation checks more than once or twice? Even then, if you're aware in advance that your character will spend a length of time bluffing, you can simply prepare the necessary spells.
Meanwhile, an adventuring day that requires the wizard to cast both Fly and Dimension Door twice outside of combat would leave the Rune Knight high and dry. Your fighter also cannot swap their loadout based on foreknowledge. A wizard that expects little combat can tailor their prepared spells to match. A "social" Rune Knight cannot exchange their build for a more combat-focused one if they know they're going to be dungeon crawling.
As an aside, you also don't have very good offensive combat spells unless your DM plays on easy mode. You have Polymorph and Web, but the rest of your offensive spells are pretty weak and you don't have a lot of options. Go up against something like a young red dragon or a Bone Devil for example and you are not going to be nearly useless in combat .... as a Wizard!
The wizard is effective at combat, not dealing at single target damage. That means disrupting enemies with Web, slowing them down with Ray of Frost, or turning an ally into a Giant Ape via Polymorph. Single target damage can be left for the classes that excel at it like a paladin, a ranger or, yes, even a decently built fighter. Not one with 12 STR and 16 DEX.
What does Tiny Hut do for the social or exploration pillars?
Do you really not see the value of a foolproof place to rest when you're out exploring?