Pushups arguably _are_ weightlifting, it's just that you're lifting your bodyweight. Pretty similar movement to a bench press, just upside down.
I like lifting "actual" weights more than I like bodyweight because it's extremely easy to quantify your increase in strength. But any kind of weighted exercise is amazingly fun and effective once you get into it.
Sample size of one, but in my younger and less careful days I have managed to injure my shoulders overdoing both push-ups and dumbbell bench press. I managed to injure my shoulders much faster doing push-ups, likely because there was no way for me to ramp up the weights slowly, since push-ups are always minimum body weight. With bench press, I injured myself when I got impatient and ramped up the weights too fast. With push-ups, I injured myself when I got greedy and did too many sets throughout the day, since it's easy to do and do twenty whenever there's a few minutes of free time.
There are various ways to modify a pushup. The absolute easiest that I know of being a standing push up -- stand, plant your hands on a wall, and do the push up motion. Horizontal pushups can also be made easier.
Feels dissatisfying to not even do a "real" pushup, but on the other hand, shoulder injuries are hell.
My body weight fluctuates from day to day and throughout the day. If I use the same weights, or well calibrated weights, the weight I'm moving is consistent every time.
I like lifting "actual" weights more than I like bodyweight because it's extremely easy to quantify your increase in strength. But any kind of weighted exercise is amazingly fun and effective once you get into it.