Bench press vs pushups?

Max, as you note, what you lose if you do bench presses rather than pushups is the full-body aspect of pushups. Pushups train core strength (front and back), because you’re essentially holding yourself in plank position, and they also require that you hold correct form in the mid and upper back around the shoulders and shoulder blades.
Does that help?
Jane

If you’re alone, the safest way to add weight is to wear a weighted backpack or use a weight vest. Or instead of adding weight, you could elevate your feet and/or do an offset-arm progression towards one-armed pushups. If you have a band, you could lay it across your back and twist an end in each hand, so the band is providing resistance as you rise. That doesn’t add effort to the lowering phase, however. If you’re working out a lot at home, it’s probably worth investing in a weight vest. I love mine.