I believe TT players should build their work out regime around these main pillars:
CORE- this should include; side planks, single arm farmers carries, bird dogs, resistance band training, and any other pertinent ISOMETRIC exercise for the core.
LEGS- squats, split squats (a** to ground), calf raises (multiple variations), hamstring work, single leg RDLs, and most importantly, PISTOL SQUATS (these take a while to work out to, but if you are interested, i can provide you with a resource that has the progressions needed to be able to do so....in 2 months you should be able to pistol squat assuming you are within a healthy body weight and otherwise do not have any other injuries). I say pistol squats are the most important because they require strength, balance, and mobility...key tenants to success in TT.
Agility and stamina- the best way to improve this is movement drills IMO because it is the most similar to playing actual TT. if you have a robot you can actually practice this while actually hitting balls. even if you don't, you can still shadow your movement drills. Ti-long has some helpful videos on his youtube channel. wind sprints and cone drills are also helpful.
MOBILITY- there are so many different mobility exercises. think of these as similar to and inclusive of yoga type movements but not limited to simply yoga movements. Look at some of the mobility exercises that soccer players do when they are warming up for a game. hip openers on a mat with a kettlebell. leg swings. knees to chest. there's hundreds of these. does a certain part of your body feel stiff or immobile? google a mobility exercise that focuses on that part and start adding it to your weekly routine.
finally, i did want to say that i believe pushups are beneficial for TT players because they strengthen both your forearm and your wrists. As others have mentioned, stronger body parts will fortify you against injuries. Take your push ups to the next level by making them burpees. This is both a strengthening and aerobic exercise. Very good for TT players!