Before talking about skills design-wise, I need to mention how much I dislike the technical state of the current FE8U skill system: sure it’s great tech, and what’s even greater is that it is very much a community achievement, but it is also very much a feature-creepy mess that I find very hard to work with and even has gameplay issues with all of the lag going on. I guess that the unfortunate state of buildfiles in general could be to blame for that. 
Anyway, skills are an interesting subject. At their core, skills are just an abstraction over abilities that each individual unit has. Anything that is not an innate property of every unit could be a skill. Whether or not making that abstraction is sound design wise is, I think, arguable. It probably depends.
First, the elephant in the room: proc skills are indeed kinda wack. They’re fun spectacle, but don’t really add much when it comes to designing tactics, and are mostly just fancy crits without much counterplay. Combat arts are probably more reasonable alternatives to them.
That being said, my problem with skills as a whole, as they are implemented in most games in the series as well as in the skill system used by many contemporary hacks, is that they add, in my opinion, a lot of unnecessary complexity. That is they add complexity without adding very many new options for the player.
They add complexity because they make the player need to investigate the skill set of each individual unit to be able to make informed tactical decisions. How much complexity is added of course depends on the complexity of the individual skills, as well as their density (see the popular “skill bloat” argument). For GBA specifically (well, really everything except 3DS which has a bottom screen), it also doesn’t help that you have to actively check a unit’s stat screen to check its skill set (vanilla games mitigate this by only giving skills mostly to units important enough that you’d want to check their stat screen anyway, such as player units or bosses).
They do not add options because each unit’s skill set is mostly fixed. This of course is not entirely true as especially vanilla games give you ways of altering skill sets for player units at least, but I do have issues with most of these methods.
- FE5 and Tellius have skill scrolls. My problem with those is the same as my problem with stat boosters, in that they invite hoarding through their fundamental design. I would much rather see something like FE4 stat villages/talks or TRS skill houses, which are basically the same thing but limited in availability (forcing the player to “use” them as they get access to them).
- 3DSFE, 3H and ig the GBA Skill System have skill learning and give you choice of which skills you learn through reclassing and branched promotions. This is fine for players experienced with the game I guess, but does not add anything for blind players, which should be your primary target when designing a game such as Fire Emblem. Blind players have no way of knowing which skills they will game via which class path without some kind of guide.
- FE4 has skill inheritance. This has similar issues to games with skill learning since its only cool when you know what’s coming. It’s also very limited I think.
- FE10 allows you to rearrange the skill set of every of your units between every chapters, as if skills were equipment pieces in an RPG. No real complains there, which is why FE10’s skill system is my favorite in the series by far.
I also really value simple, elegant design and I think skills can, often times, clash with that. Even if you’re only using stuff like Steal/Canto/Shove, why do you need to hide these abilities behind the skills abstraction at all? Why couldn’t these abilities just be innate attributes of thieves/mounted/unmounted units? You like Vantage? Why not make that a weapon effect instead. Charm? Is +2 damage/+10 hit or whatever interesting enough to warrant the added need to check for it?
I guess there probably is value in considering skills even when designing a game with simple mechanics. But I do think that it is probably better to avoid using them unless you really believe you need them in your design.
An interesting alternative to skills I kind of like in theory is something that 3H experimented with a little: accessory items. Instead of have a skill have a certain effect you could tie that effect to an accessory item which units can equip. Unlike skills, accessories can be traded freely between units and allows for interesting decisions to be made by the player (as with any item). Of course there’s no really accessible accessory framework currently available for GBAFE but that’s only a problem when you ain’t a wizard 
Hopefully that wasn’t too bad. Words are hard.