I did a hypertrophy block / mesocycle with Tate presses as an accessory after my main movements and they definitely thickened up my triceps a bit. Started at my minimum effective volume for triceps than added volume and upped intensity through the training cycle.
Then used close grip bench as assistance in the next block and made similar tricep gains tbh without the Tate Press
Conclusion: Tate presses are handy if you don’t cgb enough to fully stimulate triceps or do so much comp work that it’s limiting the weight or volume you can CGB.
Haven’t exactly had lockout issues myself. Thought I did at one time. Turns out I was just slow off the chest, not by enough to fail there but enough to make sure my lockout was always a grinder. Got tighter and more powerful off the chest and the extra momentum meant lockout was never again a problem.
Another issue I fixed up was not doing hard lockouts every rep. Got this from Brandon Lilly. I like a lot of people I’ve seen did about 90% of the ROM not fully locking about before going for another eccentric.
Look at Terron Beckham. Super strong bench presser. Super strong off the chest. Big arms. 99% of his reps he skips the top little bit of ROM. Fails lockout all the time.
Anyways point is that bigger triceps only give you the potential for greater lockout strength. To potentiate this you need to be doing bench press variations on which you lockout e.g. CGB where the increased ROM and tricep emphasis challenge lockout.