Here is some of the issues I consider needing fixing.
Combat : I think
@Cap'n Kobold covered the most important points regarding single target damage and conditions. What I think is more glaring is the fact that Martial don't tend to get NEW OPTIONS as they level up. They get better at stuff they were already doing at level 1-3. A Battlemaster's maneuver options are the same at level 3 than at level 15 and you just end up choosing stuff you rejected in previous levels instead of picking fresh new options. The same occurs in the other pillars of play. Unless you have spells, you don't get new tools.
Social: Fighter and Barbarian get pigeonholed into the intimidate skill, the WORST skill in the game. It's the one skill that tends to make things worst, regardless if you roll well or not! If you start intimidating someone,
they're not going to like you even if you're effective. And they're more likely to lie to you to make you go away!
The problem you have to overcome is the attitude that if a martial character can do it, anyone should be able to. The whole "why can't my Bard dress up as a fighter and get the same bonuses?" "Rogues shouldn't be able to be better at Arcana than Wizards!" and so on.
That problem too! Martial characters are too often seen as some sort of baseline for what a character can do, even though they already have unique abilities! People think any random Town Guard is a level 5 Fighter or something. That kind of attitudes needs to go away. Fighters aren't just any run the mill soldier or town guard and the game doesn't do a good job of making you sound like something different the nyou take the class.
The same with the idea that real world professional athlete are Epic. Epic martial characters should be BETTER than real athletes... but they're not! The maximum distance a 5e character can cover with a standing long jump is 10 feet. 11 feet if they're a Barbarian with 22 STR. Note that the game doesn't give you any penalty for your gear so it doesn't enter in the calculation. The world record for the standing long jump is a little under 12 ft 3 in. A level 20 DnD hero should be able to overcome that limit as a basic ability.
Exploration: The biggest disparity when it comes to overcoming obstacles is in narrative control. Spells are discreet options with a reliable result. You need food in the wilderness? You know what Goodberry costs and what it results in. You don't have a spell? Time to roll and pray for a good result! Martial always have to
improvise and use
creativity to do anything but the classes with spells are not only perfectly able to do the same, they get more tools to be creative with (like Mage Hand alone!). That said, the
Rogue is a great example of this with their later ability to just Take 10 whenever on certain skills. This is the sort of things other martial classes need.
I don't think it should be out of line for Fighters and Barbarians to be able to trade their Hit Dice for extraordinary effort, for exemple, or the Fighter be able to trade their Action Surge to perform an incredible feats of strength.
Especially considering how RARELY the skills given to Fighters and Barbarians types actually come into play. Especially as the game goes on and casters gain more and more reliable options.