Well, clearly they
are related to skill use they're just not, as I said, in a 1:1 relationship.
The rest spoiled for length:
Spoiler You can't get any perks for a skill if you don't use it, so there's your relationship. That doesn't mean that the relationship is as good as it can be, of course, and if the discrepancy is too great, it's a problem. I'm happy to agree with you on this. And I agree that it is illogical to train a different skill and use that experience to acquire a perk in the first skill. But I don't think that's what the developers were really going for.
I think the problem is that taking a min/maxing approach to character builds ends up inverting the relationship that player's take toward progression. If you, logically, assume that 100 in a skill = the best you can be and therefore a master of every perk available in that tree and then realize that you can't get every perk by reaching 100 in that skill because the xp allocations don't allow it, it doesn't make sense and it looks like a design flaw. But, realistically, you're not going to reach 100 in a skill in isolation. You're not swinging a sword in a vacuum. You're acquiring experience in a variety of other skills simultaneously and that other experience allows you to invest in perks. If you play the way they expect you to, and the way the game has been designed, you can achieve all of the perks in a skill tree by the time you reach 100 because you've borrowed xp from other skill trees to do it. You've invested your extra 'energy' in mastering one skill over another. In other words: they've rewarded your choice to specialize. This goes back to what I said about perks being like special tricks you can learn, but don't have to. You can spend thousands of hours on the firing range improving your aim, but that doesn't mean you're learning how to quick draw or trick shoot. Those are separate skills that someone with less experience on the range could master before you.
The 'broken' relationship between xp gains for leveling a skill and the leveling requirements for mastering all of the perks is probably not accidental. They might not have been aware of the illogical consequences of breaking out a spreadsheet and min/maxing the numbers (though they might have been) but their design focus was aimed at a different goal than yours: they wanted a system that forced players to create different builds so that not every character would be an everyman, master of all perks by the end of the game. That was a complaint against Oblivion's design that they hoped their new design would correct. People playing Oblivion complained about the very thing you're saying they broke in Skyrim. Who's right? The people who didn't like that they could max out every skill, thus eliminating the class mechanic altogether? Or the people who like to be able to master every single skill? If all it takes to get every single perk in a tree is to use only the experience gained from that skill, then every player can max every skill and acquire every perk. You might not like that design, and it certainly breaks down logically if you take a skill in isolation, but I think it's just a side-effect of their design objective (forcing different character builds without restricting player choice to class selections) and if they were even aware of it, they probably didn't consider it worth worrying about. I think the design is very clever, personally, though I don't think it's perfect. (For example, you should never be forced to use a skill, like Speech, just by selling stuff.)
The problem is, which you point out quite clearly, is that, unless you specialize early on, you end up being forced to train other skills to keep acquiring perks in a skill you have maxed. You haven't acquired all of the perks because you've been spending perks here and there as you play instead of dumping all of them into a single skill tree. This is a legitimate complaint, because you are expected to spread your perks around a bit, so their design forces you to specialize without choosing a class, but breaks down at higher levels. The design prohibits you from creating a character that is both a Jack-of-all-trades and a Master-of-all-trades. Is there a way to force specialization on a player without creating this problem? Should they force players to specialize if they don't want to, or allow them to acquire every perk in the game? I believe, as I said, that they did it to address complaints that players made about Oblivion, but maybe they should have ignored those complaints and left it the way it was? You can't design a system that makes everyone happy because, by definition, everyone wants different things.
It's not just a "power-gamer" mentality that would have a problem with this, however.
In Morrowind and Oblivion, both, I enjoyed the fact that I could change out how I played the game dynamically. I started Oblivion, for example, with a sort of "Nightblade" build using bows and light armor and magic. It turned out I really couldn't use bows terribly well, however, so I switched to blades. Most of the time, at least. I still used it for long-bomb sniping, especially with poisons. I fought with a dagger and "drain" spells when it came time to melee, though.
What about perks, though? It punishes not-specializing. If I put perks into a skill, then decided I liked some other playstyle better, I can't get my perk points back. As someone mentioned before in another thread, it functionally forces the sort of stereotyped uses of the skills and archetypes while the lines are generally much more blurred in the non-Arena games. A perk I have spent on archery is a perk I'm never getting back.
More, perks are devastatingly effective one-shot boosts to the power of skills. It heavily rewards pumping up one skill you use all the time by the fact that any skill that doesn't have the benefit of perks is massively underpowered by comparison. (Consider the difference between destruction with and without perks - you have spells that cost over twice as much, deal 2/3s as much damage, and lack the stunlock component or the instant disintegration. There's just no way to rely on destruction without the perks.)
Sure, some people have tried to come up with alternate solutions - that we could un-allocate perks or else gain extra perks through killing extra dragon souls, but this goes beyond the point. I actually rather dislike the whole notion that we even have to plan our growth out at all. (This gets us into the whole debate about "power-gaming" in the first place - the perks CREATE power-gaming by the fact that you have to plan your growth and have obvious better-and-worse perks to choose. Or at least, it creates a new way to power-game, there certainly were ways before.)
As one of those things I want to do when I get the CK, I want to actually break this perk progression off from levels, and make it based upon an additional set of sub-skill experience. Rather than having lump-sum purchases with perks, you simply focus your "experience" gain with a skill towards gaining some particular within-the-skill specialization.
This can be done either through manual selection (which may be necessary, but not preferable), such as just picking a line of perks you want to learn first from the skills screen, and as you gain skill experience, you naturally start unlocking those perks. Better, this could go without the lump-sum nonsense of suddenly having your armor protect you 20% more than it did just a second ago right in the middle of combat. You could simply have that modifier have much more subtle gradients of .1% bonus defense each time you get to a minor milestone. Because we aren't using one single pool of perk points, we have no need to try to make all perks cost the same and be balanced entirely to one another.
The alternative, which would be better for immersion, but may not necessarily be possible in many cases, would be a natural extension of growth-through-use - just make your perks come from the manner in which you use your skill. You get the fire damage bonuses as a sub-skill specialization of using fire type spells in the Destruction school. The more heavily you favor the fire spells, the more they will shine in comparison to the other types of damage in the school. No planning, no power-gaming, no forced specialization, just let the skill develop purely from how you play.
As for "you should need someone to teach you to quick shoot", I think this is fair. I would actually make you have to get training, and possibly perform quests to get that training before you can learn those perks. (In sub-skill skill training, this would mean that it would let you start putting skill xp into those perks.) It would make the trainers actually have a little more purpose. If you were dead-set on keeping specialization in the game, you could easily make some quests mutually exclusive, so that you cannot learn every perk in the game, at least without further modding.