Couldn't view the video unfortunately.
However, just based on what you describe your best bet is working on your technique and also doing a lot of benching.
So, step one is to find the most optimal technique for you: grip width, flat/raised heel, setting up by pulling through or feet on the bench, etc.
Step two is to groove that technique with plenty of volume. Make sure most of it is light enough to allow proper execution but heavy to need effort. I've found around 75% for sets of five work well to start with. Add in some heavier sets but focus on those grooving sets and try to do them every day you train except heavy bench day.
Beyond that, without seeing the video (won't come up on my phone) I can't give you specifics.
Final note, with the back tightness it probably won't hurt to do more upper and middle back work.