First, I've never watched the Dave Tate video...Second, I'm by no means a pro and my bench press is the worst of my lifts, I've always struggled with it but recently I've made some good gains.
I watched the Westside Barbell video tapes probably 15 years ago or better and Louie discusses bar path, pressing in a straight line. The easiest way I've found to implement this is to bring the bar to your sternum, it will almost automatically force you to tuck your elbows and push the bar in a straight line. If you're currently benching like a bodybuilder (across your pecs) it's going to take some getting used to.
Next, one of the huge things that is tough to really explain is squeezing your shoulder blades together. It's almost like when people do shrugs and they rotate their shoulders backwards...when you do that, it shortens the "throw" of your arms.
Next, don't neglect training your lats/upper back...I did this for so long because I never seen how it could help my bench and I suffered for it.
IMO, things like skull crushers and pushdowns are of limited value to building a big bench. If you use boards, you will be able to handle much more weight than you will ever be able to do with a skull crusher...I still do them occasionally but boards are superior in my opinion.
Overhead presses have done great things for the guys I've been training with...
that's about all I've got.