I think the best way to do the skill system would be to have skills with subskills beneath them. Like, for example, the one-handed tree (which I think makes more sense as a seperation than blade and blunt) would have blunt and blade (or something like that) as subsets, and have some perks in those subsets. The total blade and blunt level average is equal to your one-handed average. Levelling up is done by having a certain number of points go to the subskill you're using, and a smaller number of points going to the other subskill(s), so this way, practicing with a sword makes you still improve your ability to use a mace, but if you use a mace more often, you're ability to use a mace will improve more than your ability to use a sword. Some skills may have more subskills than others, but it doesn't matter as your skill revolves around the average. Additionally, a revamped attribute system would serve as another umbrella, like with strength, so this way, improving your skill with mace or sword also improves your ability with hand-to-hand or greatsword or warhammer. This is for the simple fact that stronger people hit harder with EVERYTHING, not just a particular weapon they've done the most training in. Other skill subsets could include light and heavy armour, speechcraft and mercantile etc.
Perks COULD be a good thing, but not the way they were implemented in Skyrim. Instead of saying "Lets make it lyk a cool constellation, lolz" let's say "Let's organise the perks in a logical manner". Perks need not be in visual trees within skills. Perks can have prerequisite perks, and prerequisite skills, but they need not have only one of these, and it should just be done to make sense. For example, if you want a perk that does higher sneak damage with a dagger, you should have a certain prerequisite level of both sneak and that weapon. Skills are the main focus, attributes are for generalisation, subskills and possibly perks are for specialistion.
I've heard a few people say that simplifying the skill system so there's less numbers makes more sense, but that is complete rubbish. People don't have 18 skills. People have complex forms of abilities and simplifying them just makes them seem like awkward robots with lapses in evolutionary logic. Someone who's good at sneaking stealthily will be better at stealing than someone who isn't. Someone who does one kind of magic has more magical knowledge and experience than someone who hasn't, even if their experience is needed to be applied to a different magical field. Yes, we know the numbers are not realistic, because you don't see numbers in real life. They are a representation of the detail we can't see, but which is subtly working it's effect. After all, a game's representation of reality is limited, if you want to do a complex skill system, you need numbers. Visual aids are good too, but not so much obvious ones that it seems artificial.
Here is my ideal skill list, I think. I think the 18 skills system of Skyrim would have worked fine, if it were not that they were intrinsically seperate with the perks failing to offer logical specialisation. In case you're wondering, I did not include athletics as I thought it would be too much grinding. Rather, I think speed should be goverened by your stamina, or possibly an attribute.
One-handed
Two-handed
Defense (Includes all armour and block)
Marksman
Hand-to-hand (utilised to include fist weapons like katars, claws, brass knuckles etc. as well as things like kicking and special moves)
Smithing (Includes repairing)
Sneak
Security (Includes lockpicking and stting up traps and other similar things)
Guile (Includes speech and mercantile)
Alchemy
Acrobatics (Includes parkour moves and dodging in battle)
Theivery (Pickpocketing, which can also be done by distracting people while you walk past them, and how easily people are alerted when you take something)
Destruction
Alteration
Restoration
Illusion
Conjuration
Enchanting
Here's the list again with my current thoughts for subskills liste, soilered to save space.
Spoiler
One-handed: Sword, mace, axe, dagger
Two-handed: Greatsword, warhammer, battleaxe, spear
Defense: Armour (I really don't think it should be seperated by type, it's a meaningless division), block
Marksman: Archery, throwing weapons
Hand-to-hand: Unarmed, hand-to-hand weapons
Smithing: Don't know
Sneak: Avoiding detection, sneak attacks
Security: Lockpicking, trap disarming and setting up
Guile: Persuasion, mercantile
Alchemy: Don't know
Acrobatics: Climbing, dodging, jumping/falling
Theivery: Pickpocketing, stealing (from buildings as opposed to people)
Destruction: Fire, frost, shock, direct effects
Alteration: Don't know
Restoration: Don't know
Illusion: Don't know
Conjuration: Summoning, necromancy
Enchanting: Don't know
Two-handed: Greatsword, warhammer, battleaxe, spear
Defense: Armour (I really don't think it should be seperated by type, it's a meaningless division), block
Marksman: Archery, throwing weapons
Hand-to-hand: Unarmed, hand-to-hand weapons
Smithing: Don't know
Sneak: Avoiding detection, sneak attacks
Security: Lockpicking, trap disarming and setting up
Guile: Persuasion, mercantile
Alchemy: Don't know
Acrobatics: Climbing, dodging, jumping/falling
Theivery: Pickpocketing, stealing (from buildings as opposed to people)
Destruction: Fire, frost, shock, direct effects
Alteration: Don't know
Restoration: Don't know
Illusion: Don't know
Conjuration: Summoning, necromancy
Enchanting: Don't know