There are many ways to gain additional skill proficiencies in the rules.
If you or your players want to make a new set of proficiencies, you're free to do so using a customized background: (PHB 125)
You might want to tweak some of the features of a background so it better fits your character or the campaign setting. To customize a background, you can replace one feature with any other one, choose any two skills, and choose a total of two tool proficiencies or languages from the sample backgrounds.
The Skilled feat (PHB 170) is self-explanatory:
You gain proficiency in any combination of three skills or tools of your choice.
As a DM, you can let your players get training in skills in order to gain proficiency: (DMG 231)
A character might be offered special training in lieu of a financial reward. This kind of training isn't widely available and thus is highly desirable ... Possible training benefits include ...The character gains proficiency in a skill.
However, consider that this bonus is in the same list as gaining an extra feat, so it's not something to be done lightly.
Finally, both the rogue and the bard have ways of boosting their skills. Rogues can double their proficiency bonus for a handful of skills through the Expertise feature (PHB 96) and bards get half of their proficiency bonus to any check that doesn't include their proficiency bonus (PHB 54).
You might be interested in the Background Proficiency variant rule
DMG 264 has the "Background Proficiency" variant rule, which seems to hew pretty closely to the mechanics you desire:
With this variant rule, characters don't have skill or tool proficiencies. Anything that would grant the character a skill or tool proficiency provides no benefit. Instead, a character can add his or her proficiency bonus to any ability check to which the character's prior training and experience (reflected in the character's background) reasonably applies. The DM is the ultimate judge of whether the character's background applies.
You don't need specific rules to make crazy situations
Why did the designers get rid of the skill point system in the first place? Mike Mearls said:
We unified the progression for skills and weapons under one set of rules. That removed a lot of complexity and allowed us to include skills in the simplest version of the game without adding a lot of rules overhead.
Therefore, instead of depending on an arguably unwieldy and complicated set of rules for skills, you can use the simplified framework of 5e to create the kinds of specific "proficiencies" that you describe in your question.
More specifically, you can let your players add their proficiency bonus to specific kinds of checks, or even give advantage for more significant bonuses. If your character is an expert in astrology, maybe let them add their proficiency bonus to Arcana, Nature, or Religion checks that involve the stars. Examples of this kind of bonus exist throughout the system: for instance, the Belt of Dwarvenkind gives advantage on persuasion checks against dwarves.
Indeed, such cases are what the advantage mechanic was made for: the DM giving quick bonuses in response to favorable situations. You might find that this goes a lot smoother than a skill point based system. As a player, I've found that DM bonuses like this are not only balanced (because they're highly situational), but also a lot easier to keep track of than, say, 3.5's super complicated skill system.
These are approximately equivalent.
2 Skill proficiencies are more useful than a language overall (both in my experience and according to the rulebook in so far as you can train languages, but not skills, with the Downtime rules in the Player's Handbook), so you have to be careful, but losing your choice of a cantrip for one of the least useful ones (in my experience) should account for this discrepancy.
This can also be seen by comparing the Half-Elf and High Elf racial traits side by side (as these official races are roughly balanced with each other).
">" is slightly better than; ">>" is noticeably better than; ">>>" is much better than
\begin{array}{rcl}
\bf\text{High Elf} & & \bf\text{Half-Elf} \\
\hline
\text{+2/+1 ASI} & \text{<<} & \text{+2/+1/+1 ASI}\\ \hline
\text{Fey Ancestry} & \text{=} & \text{Fey Ancestry} \\ \hline
\text{Darkvision} & \text{=} & \text{Darkvision}\\ \hline
\text{Extra Language} & \text{=} & \text{Included}\\ \hline
\text{Keen Senses} & \text{≈} & \text{1/2 Skill Versatility}\\ \hline
\text{Cantrip} & \text{>} & \text{1/2 Skill Versatility}\\ \hline
\text{Trance} & \text{>} & \text{none}\\ \hline
\text{Elf Weapon Training} & \text{≈} & \text{none}\\ \hline
\end{array}
I consider Keen Senses and "skill of your choice" to be roughly equivalent since Perception is one of the more useful skills available from experience and WoTC Adventure Module analysis. Elf Weapon Training is a very minuscule benefit.
Now let's take your Sun Elf Homebrew and compare it in the same manner.
\begin{array}{rcl}
\bf\text{Sun Elf} & & \bf\text{Half-Elf} \\
\hline
\text{+2/+1 ASI} & \text{<<} & \text{+2/+1/+1 ASI}\\ \hline
\text{Fey Ancestry} & \text{=} & \text{Fey Ancestry} \\ \hline
\text{none} & \text{<} & \text{Darkvision}\\ \hline
\text{none} & \text{≈} & \text{+1 Language}\\ \hline
\text{Noble Heritage} & \text{<} & \text{Skill Versatility}\\ \hline
\text{knows } light & \text{>} & \text{none}\\ \hline
\text{Trance} & \text{>} & \text{none}\\ \hline
\text{Elf Weapon Training} & \text{≈} & \text{none}\\ \hline
\text{Celestial Resistance} & \text{>>} & \text{none}\\ \hline
\end{array}
This hinges on the fact that Necrotic and Radiant are rarer damage types in your setting and that History and Persuasion are not used more often in your campaign. In a setting with heavy intrigue and/or many creatures that deal Necrotic or Radiant damage, this race may be unbalanced as Celestial Resistance is pushed to much better than status and/or Noble Heritage is roughly equivalent to Skill Versatility (I've not done any analysis on the proportion of campaign-styles where this is relevant like the WoTC balance team likely did, so I can't say for sure if this is balanced overall, but it seems to be).
Best Answer
This will imbalance the party
Even with your changes, I would not expect Barbarians or Paladins putting lots of ASIs into Int, so in the end you will just make the already most versatile Wizard even more versatile.
Easy fix
Stop giving rewards after Int 14, the only people above it (Wizards) don't need extra motivation.