Honestly it was the the not so tacit, Fighters are Unskilled that annoys me.
Everyone? Multi-classing being seen as a must have for someone interested in having a decent skill set.... hmmm reminds me of a slightly subtle version of arguments that looked like
"Don't like how we let the fighter fade as they leveled and minimised his decisions in combat, you really want to be an X".... ahem back in the day that was basically the argument as framed when we pointed out the ones who might really be most interested in combat ended up having the least choices there-in and eventually the least power.
4e did a way with that and really tried to level up the playing field at least in combat succeeded they made some headway out of combat too so i am probably spoiled LOL as Rituals roped in utility magic a lot and Martial practices were an interesting idea that almost...
Skills really are the utility abilities of the game.
Although admittedly the 4e multiclassing doesnt necessarily do a super Schwing to the character flavor Knights were commonly hunters too so your suggestion is not super terrible for flavor either
Typically 4 (5 only for human or Eladrin fighters right? Humans are on my favorites list for some reason)
The background particularly the ones from Scales of War may open up the selection set as already mentioned
but has no impact on trained number of skills.
What I have heard of 5e has yet to convince me of much, but I am open to changing my mind, I saw some interesting design creativity in the 5e precursors.
Yeah, I think the "everyone should MC" thing was an unforeseen effect, but its true, MC feats are just too good to pass up, though of course you CAN pass up anything in 4e and it won't really HURT you... But I think that the flavor is OK too, as you say, and you don't have to frame it as "now I'm a ranger" or whatever, you can just frame it as you learned some tricks, reflavor them however you want.
5e parses the skill/class/subclass/theme space a bit more systematically than 4e. 4e has a pretty good class lineup, but then they got a bit crazy with it later on, and there was never a really STRONG subclass concept, which would have been nice, and which 5e does have. Themes in 4e are nice, but they tend to just clutter the game, especially when you have backgrounds as well, and Power of Skill, MC feats, etc all chipping in more 'stuff'.
I think a good redesign of 4e would try to push a lot more build concepts on fewer classes and do it with less granular choices, so to make 'axe wielding melee dwarf' would be pretty much one class, one race, and one option, like it is in 5e. In 4e you have to keep picking more different feats and etc to really manage your concept. It can get a little more intricate than many players want to bother with.
So for instance in my 4e campaigns I've ALWAYS had one 'gearhead' player (not always the same player) that would deal with character builds and advise the others and find them "some feat that works to make me better at X". Sometimes I had to fill that role myself as DM, but often one of the players would step up, thankfully, and pick up a copy of DDI, code in all the PCs, and print them all, level them all, etc. I never had a whole group where everyone would do their own. Back in the 2e days that would never happen. I noticed that mostly in 5e people could all handle it (even my mother picked out what her elf rogue did, and she's 80!).