Yes, 'core strength' is really important, and it isn't the same as 'abs', because the core is deep musculature including the diaphram, the pelvic floor and the deep abdominal muscle called the transverse abdominus. Air pressure is an important aspect of 'core strength' as well, called 'intra abdominal pressure'.
The exercise you describe is usually called a 'burpee'
https://www.youtube.com/watch?v=dZgVxmf6jkA. Of course it's very effective across cardio vascular and muscle endurance aspects. It's very popular with Crossfit athletes.
Where the benchpress is concerned, it is quite a technical movement involving the whole body, but probably too technical for this thread. Pushups are excellent, the best front upper body drill of all, really.
In this age when people are always hunching over their phones, computers, desks, steering wheels and so on, back exercises have become really important in exercise routines. So pull up, high rows, shrugs, and rear shoulders... pulling motions.
We usually work complete with upper body/lower body - push/pull.
The routine you describe is all push so one will see development in front portion, but not in back, which in practice would exacerbate the hunched position common to the typical modern man/woman.
The lower body isn't designed for pulling, just pushing so we don't have 'pull exercises' for legs and glutes per-se.