Arcana is not spell casting or magic at all. That's why there's no check to cast a spell and a character doesn't learn to cast spells in having proficiency in arcana. Extra proficiency is also not a wizard (or cleric) field. Casting spells are their fields.

The skills have nothing to do with spells or spell casting at all to the point arcana, religion, nature, etc are even required to be taken by those respective classes. If a bard wants to take expertise in arcana the only thing it does is better equips him to recall more rare lore.

Since variations of bardic knowledge have existed in every single edition that shouldn't be surprising at all. Rare knowledge is definitely a bard thing stemming from it's celtic inspiration in the fili.

Rogues don't have arcana, history, nature, or religion on the class list at all. They are not taking expertise in the rogue class abilities for arcana. They are taking expertise in the background. An sage with expertise in arcana should not be surprising either.

Each of the recalling lore skills is the ability to recall the lore, not the actual technical training in having learned the lore in the first place and the role is only relevant to recalling higher degrees of rarity. In the arcana example a wizard does not need to know the mating rituals of creatures that live in the Abyss, which is what an extremely high arcana check (for creatures in the outer planes) might give, in order to cast magic fireball.

The wizard focusing on INT is what gives the wizard that fluff as booksmarteducated. The baseline in 5e is the d20 with ability score modifications. People look at the highest totals as the goal for some reason when it's overkill. Most DC's are 10 or 15, and most of the time rolls are not necessary.

A character is good at a check with either proficiency or a high ability score and capable of moderate difficulty under time and duress constraints. A character is great with both a high ability score and proficiency. This is readily available in the manuals, which define 10, 15, and 20 DC as the only DC's a DM would regularly assign to represent easy, moderate, or hard challenges. The expectation is a difficult challenge under combat conditions or other threats will be successful more often than not by such a character.

A wizard with a +5 INT and +4 proficiency bonus will succeed on DC 20 checks 50% of the time assuming the check even needs to be made. That's the actual benchmark for extremely educated on the subject based on how 5e handles ability checks. It's not "a higher number possible for a bard so it should be for me too", which is rather inaccurate anyway since the bard (or rogue with a suitable background) would also need that INT investment. The end result is not that different and wizards are moderately knowledgeable on everything.

Clerics and druids do not demonstrate any advanced knowledge to do their jobs. They need to know the spells and typical (ie DC 10 checks) lore. Proficiency represents additional focus and training according to the DMG on when to apply it. They can add that additional focus beyond the typical information by taking proficiency. That's above the baseline standard and neither class studies advanced lore just because of an association with that lore.

However, if a person really wants their "educated" casters to be experts in an associated skill I would use this house rule for expertise:

Bard = history
Cleric = religion
Druid = nature
Wizard = arcana

Balance that out by making it optional for the PC and removing both class skill choices for clerics, druids, and wizards (one is used for the proficiency and one is used for the expertise in this optional house rule), and give the bard performance proficiency in addition to expertise history (same reasoning but bards start with an extra skill proficiency). Allow the bard or rogue to select a skill proficiency instead of expertise when the option is available. And/or make the prodigy feat non-race restricted.

Sorry for the long post.