Though I'm honestly not sure what "skill" str could really benefit.
Strength in 3.X improves swim, climb, and jump. It also improves your ability to break and smash things ("hammering", as you put it).
Some additional suggestions (I actually use):
Porter: Increase your capacity to lift and carry things without hindrance through practice.
Run: Increase your base movement rate through practice, treating 'fast movement' as an improvable skill rather than a class ability.
You could probably also treat the ability to break things as a skill you could improve in.
I'd note that enforcing encumbrance tends to make some spell casters somewhat MAD as well. For example, one of my players created a cleric with 8 strength as his first character, only to discover that wearing any armor at all was often problematic. So his replacement character after the first died had a 13 strength, largely to ensure that carrying capacity was high enough to allow armor to be comfortably worn. As a consequence though, CON has had to suffer somewhat which makes the character fragile in other ways.
Really, unless you can fly with excellent maneuverability all the time, it's very hard to avoid mundane threats to mobility - slippery slopes, water, mud, etc. It's even easier to counter that threat to game balance than it is to buff mundane classes up to balance. Simply reducing the practicality of the flight spell in terms of speed, maneuverability rating, and duration tends to eliminate it as a win button. For example, in my game the 3rd level version of 'fly' gives you winged flight at speed 60 with an average maneuverability rating. You can't hover, you can't fly vertically up a shaft, you need a concentration check to cast a spell with an 'S' component, and you can't fly at all if you have lost 2/3rds of your hit points. The 4th level version looks like the SRD fly, except that there isn't a free feather fall when it is dispelled, so make sure you have a spell slot open in the event you get hit by targeted dispel magic. And so forth. And it's also worth noting that if you do fly, you actually pick up mobility hazards that you don't have if you are on the ground. By the rules, wind is a much bigger hazard to a flying creature than it is to one using land based movement.
Meanwhile, I've got a player of a multi-classed rogue that can bounce around like Yoda in the prequels and basically can move up and along walls or leap obstacles at will and at high speed.
So, if you are actually using the environment as a consistent hazard, yes, spell-casters will have more powerful "answers" to any particular challenge, but they do so at the cost of eating into their power. For example, in my current campaign the PC's are on a boat in the semi-tropics (soon to be the tropics) in the summer, with day time highs above 95F and very high humidity. I'm using my own rules on heat exposure and applying it daily in the form of 'non-lethal damage' that reduces each characters maximum hit points. Now, they have a cleric that can cast 'endure elements' which gives a 100% immunity to this penalty, but what they've decided to do is to only use this spell slot to protect... the three spell-casters. The three martial characters have high constitutions, high hit points, the Endurance feat, 5 or more ranks in survival and so forth that tend to mean that the heat exhaustion penalties for them are tolerable - 2-5 points of damage per day out of 50 or 90 hit points isn't a big deal. By comparison, some of the spellcasters were taking 6-12 points of damage out of 40 hit points. So, again, yes, the spellcaster has the better answer and yes it's a huge resource to the party that Endure Elements is available, but... that's three less spell-slots available for other needs. At some point the ability to act as party support is being compromised by the consistent need to support yourself as the more delicate and less skillful member of the party.
Granted, this situation is far more slanted toward the spellcasters in 3.X with its more powerful spells, its lack of protection for martial silos, and its extreme and unwarranted conservative approach to how useful skills are or how skillful you can be if you aren't a spellcaster. The "only spellcasters are allowed to have good things" problem is real if you are going by RAW. But I don't agree that is the only approach, and it appears 5e is trying to correct it.
