Should someone who is trying to get stronger do weighted carries and single leg work?If not, why?
Sure. You can get stronger doing those.
If your goal is to improve your single leg strength and weighted carry, they are the ideal exercise. Single leg exercises can also improve squat strength. Squats are best to improve squats, but single leg exercises do carryover and improve the squat. (Principle of specificity).
Strength is specific. Is your goal to get better at single let work and weighted carries?
The more untrained you are the more doing anything will improve your ability to do anything, roughly speaking. If you’re a novice I would recommend sampling many different movement patterns, such as the ones you mentioned. However, all of this changes the more trained you become. If you’re looking to improve your strength in your powerlifts, for example, you’ll need to make your training more resemble that, and doing things vastly unrelated to that are not going to provide much direct benefit.
I think that getting good at multiple movements sounds appealing. Any movement that falls under the six fundamental movement patterns.
I don’t powerlift or play a sport. So for me, loaded carries are especially important as they train a form of strength that might get used outside the gym.
For the same reason, I really like bodyweight movements, deadlifts and OHP – I might have a use that strength at some point.
Squat and bench seem less relevant, but I do them for hypertrophy and because they are benchmarks for comparing myself to others.