Carrying stuff matters less when you have things that exist specifically to carry things for you... like a vehicle. Even then, it doesn't seem like many skills you'd imagine being conventionally strength-related are actually strength related. Climbing, Archery, Martial Arts, Melee Weapons, etc. don't factor in skill at all.
Attributes only matter at 7+ for linked bonuses, for the amount of experience to go from a 4 to a 7, I could just purchase a 7 in a particular skill, gaining seven times the benefit.
Relying on a vehicle to carry things for your character depends on how evil your GM is for how good of an idea it is.
At the very least you'll want to plan out what your character carries while on foot.
While Strength may not be a link attribute to those skills it can have some pretty important effects, like determining how much damage you do, how fast you can climb*, and thanks to the aforementioned direct attribute checks can cover certain fuzzy areas in the use of those skills.
Where the XP break even point for where it becomes better to get an attribute bonus versus using the XP to increase skills does depend on what level of skills you want and how many you want that can benefit from an attribute bonus.
There are also considerations as mentioned of attribute minimums to do certain things. Inner Sphere Battle Armor training, Special Pilot Abilities, certain modules, and maybe even character concepts can all demand minimum attribute levels.
Also worth mentioning is that untrained skills are attribute checks in A Time of War, with a penalty I should mention, so that can be an incentive to invest in attributes beyond certain points too.
*Which reminds me, how far your character can walk, run, swim, crawl, and climb in a turn are linked to certain attributes. So if you want to move farther in a turn you can invest in the relevant attributes and get a boost to all those ratings while certain skills only improve certain movement rates.