On the bike to get proficient, to get better and to enjoy it more as a cyclist requires thinking about the bio-mechanical motion of the spin, the technique called spinning or ankling.
When in the drops working on anterior rotation of the pelvis to flatten the back, during a power interval, tempo or steady state riding in the drops or on the hoods for example. When climbing or sprinting working on steady relaxed breathing, using a push/pull technique, not “push ups”.
Push-ups are awesome, but should be done off the bike.
Training on the bike will build a huge engine for VO2max/ threshold power, core stability, leg speed/strength and endurance. And when fitted properly, saves the knees from the extreme loading created from the impact of jumping and running movements.