Should fighters be skill monkeys?
More like skill gorillas.

But, really, fighting is a skill, a discipline, really, that encompasses many skills, not all of them physical.
Trying to build a duelist out of a fighter class, for instance, has always been something of a challenge. He shouldn't just have high DEX/STR and BAB, he should have keen perceptions, be able to read and deceive opponents with consummate ease, be able to /avoid/ fights when he doesn't want them through deft social maneuvering, his remarkable grace, dexterity, and agility should see him through all sorts of other challenges. But, no. 2 skill points/level, or have a couple skills from a starkly limited list. Oh, and if you don't pick STR or DEX and max it to the hilt, you will suck, so forget about a good WIS to read enemies or a good CHA to deceive them.
That said, the 5e fighter is an incremental improvement. It does get a less restrictive list than ever before. It does at least get to seamlessly choose STR or DEX without paying a feat tax.
I've always wondered why traditionally fighters seem to end up with less skill points/proficiencies than casters.
I'm on a tear about this from other threads: It's all the fault of the original Thief in Greyhawk Supplement II, c1975.
What if a fighter started with 3 proficiencies and gained expertise in one at 3rd level? Would it break the game? Wpuld it trip over a rpgyes niche too much?
The fighter could subsume the Rogue (or vice versa) with no balance issues. The result would be a class that could cover more archetypes & genre precedents than both the classes considered separately (or with MCing, for that matter).
To a large extent, the reason Fighters are limited in skills is to avoid stepping on the Rogue's toes - they've been the skill monkey pretty much since they were introduced.
A part of me thinks that actually all the non-caster classes could usefully be combined into a single class (Hero, maybe), with the fighting abilities and the skills that that implies.
Thanks for beating me to it, and being more succinct into the bargain.