On your particular case, I don't think it is game-breaking to allow multi-classing at INT 12.
Since you are considering multiclassing for roleplay purposes, it makes sense to allow your character to get one Wizard level if you have INT 12. I don't see any particular issues with allowing a single level multiclass using the conditions you proposed. Your DM may rule that you cannot get further levels with only INT 12, since your Intelligence will stand below the average of a Wizard.
You might want to consider the Magic Initiate and Ritual Caster feats from the Player's Handbook (pages 168 & 169) as options to fulfill your roleplay as a "student of magic". Remember that you need at least INT or WIS 13 for the Ritual Caster feat.
1. Communicate your expectations with the players out of game
You and your players seem to have different expectations for how their character sheet should influence their roleplay. To you, it's immersion-breaking when a character acts differently than what their ability score suggests. To others, ability scores are just numbers that define mechanics and statistical outcomes, and don't directly restrict roleplay style.
It would benefit your campaign if you held an out-of-game discussion to directly address your roleplaying expectations and concerns. Discuss how you think characters should behave if their Intelligence, Wisdom, or Charisma is notably high or low. Get feedback. Make sure your players' preferences are in sync with yours.
Note that it may be an uphill battle if your players disagree. While a PC's ability scores affect what tasks they can do and their likelihood of success, they don't dictate roleplay or how the player should act out the character's thoughts and choices. Strictly speaking, it's the player's choice whether or not the PC's mental ability scores affect their roleplay, and to what degree. Assuming they wrote the character, they probably know best how the character would act and behave.
2. Use checks when players roleplay differently from their ability scores
If you want the ability scores to matter more, then include more dice-based challenges. This is where you have the system mechanics on your side. Characters with low WIS are less likely to succeed an WIS-demanding task. You can also prompt players to make checks, and then you can intervene regarding their character based on their success or failure on the check.
For example, if a player forgets or misses a clue but their character has high INT, have them make an INT check. On a success, they get a reminder or hint from the GM.
These checks can also be used (albeit sparingly) when players roleplay above their PC's capabilities. If the low-CHA character delivers an eloquent speech, ask them to make a CHA check or related skill. On a failure, you narrate that the audience wasn't particularly moved.
When intervening this way, make sure to use the “Yes, and” style of narration. Yes, the high-WIS druid walks toward the spider cave, and then has a flash of insight and realizes it's a bad idea. Yes, the low-CHA barbarian tries to speak politely and articulately, and the duke dislikes their untidy appearance and finds the speech unsettling. This way you still respect player agency, while maintaining narrative flow and nudging them to act more in-character.
3. Give in-game rewards for good roleplaying
Another way to encourage players is to give in-game incentives. Avoid big rewards like bonus XP, because that can seem unfair and subjective; from my experience, GMs who give roleplaying XP may seem biased toward certain players, which quickly results in player frustration.
Also avoid punishing players for what you consider bad roleplaying. Accusing players of bad roleplay when they are putting effort into roleplay could discourage them from roleplaying at all.
Instead, reward players for roleplaying in character and maintaining immersion. Use small rewards, like a temporary circumstance bonus for roleplaying in accordance with the ability score. This could encourage players to roleplay having low stats. They're already disadvantaged mechanically, and a temporary bonus would increase their odds of success.
Best Answer
Short answer: Dexterity is too powerful while Wisdom and Charisma are more difficult to justify changing. You can homebrew them just as you can homebrew anything, but you might be violating balance concerns for Dexterity if you do.
Okay, I admit that wasn't very short, but it is a complex answer. I'll try to tackle them more in-depth now.
First, why aren't those items in play?
The Strength- and Constitution-setting items seem fairly straightforward, increasing the strength and endurance of the body. Intelligence also fits this, being the closest representation to pure brain power. Dexterity and Wisdom don't seem to fit this mold as well because they are about the flexibility (physical or mental) to use the power that one does have. Charisma could probably work just fine, being like the Blarney stone or similar that causes someone to just have a forceful personality. However, there is justification that Dex and Wis could also be used as some creatures are inherently more dexterous or wise than others.
Second, can you homebrew them.
Same as my short answer, a DM can homebrew anything, but it raises concern with...
Third, is it balanced to do so?
This is a tough answer, and the answer is dependent on different campaigns. Let's break down the current abilities for a bit...
Strength items. Strength is something that is useful for everyone (more carrying capacity, mainly), but often is only needed to a certain point. Those who would use Strength as a primary stat likely have it higher or the same already and attacking with Strength is less-than stellar for most other builds as they can attack with their main stat (spells, Dex attacks, basically anything except Con gets offensive techniques), basically only being useful for spellcasters who are stuck in melee, need to damage, and don't have any saving throw cantrips. It can be used for AC, but you only need 15 for that and most who would desire that already are striving for it. Out of combat, Strength is very useful for hauling stuff, increased jumping, and a variety of ability checks. This is really the pillar where the gauntlets shine, but the effect isn't tremendous. The most important part is the fact that the wizard who dumped Strength no longer has to worry about failing DC 5 checks, mitigating a great amount of danger that would be ridiculous to fail on.
Intelligence is an odd choice in that it has precious few combat applications (except for wizards!), but is also fantastic for exploration through its number of knowledge skills and Investigation.
Constitution is the opposite of Intelligence, with the vast majority of its improvements part of combat. HP and Con saves (including concentration) can play a great part in survival, while Con checks have sparse but important uses (succeeding on Dash checks in chases, not passing out from liquor, etc.).
From the above we can see the three paradigms.
Intelligence mainly favors out-of-combat applications and is uncommon, Constitution mainly favors combat and is rare, while Strength has a little of both but not enough to get past uncommon.
For the more versatile abilities ...
Charisma becomes fairly easy to judge. It is to the social pillar what Intelligence is to the exploration, extremely useful in that sphere but limited in-combat. Thus, the Charisma item could safely be set as uncommon.
Wisdom is harder. It has just as many skills as Intelligence (5, more than Charisma), including one of the most important for exploration and combat (Perception) and the most important non-Charisma skill for social (Insight). Already, it is riding up in rare territory due to being useful for skills in every pillar. Further, it is a major saving throw (one of the key reasons why Constitution is a rare item) that corresponds to debilitating effects that can outright remove the user from combat. That is a really potent ability. I would say Wisdom is at least very rare, it is just that powerful.
Dexterity is similar to Wisdom. Stealth is incredibly useful for both exploration and combat, Acrobatics provides one of the primary reasons why non-Strength users use the gauntlets (escaping grapples and the like), and Sleight of Hand can play a key role in both exploration and social pillars. Further, it can be used to increase the weapon capabilities of non-primary users, another key point of Strength. We are already a solid rare, maybe getting into very rare, as it is an ability that takes most of the powers of Strength and then powerful uses in everything in other ways. Finally, it also gives AC, but this one is even more worrisome than Strength. With that ability, you needed armor proficiency to really make use of it, but Dexterity can be used for AC by everyone. Barbarians can get a dramatic rise out of it if they have been focusing in Con, while both wizards and sorcerers can suddenly find ASIs able to be devoted to feats like War Caster rather than buffing AC. Even strength-users will love having a decent AC when they are caught otherwise naked! On top of all this, it is also a major save (admittedly more for damage than dangerous effects). Definitely very rare, maybe getting to legendary.