I think the strict press is pretty important for the reasons that Pwnisher already laid out and for the fact that when it comes to repping something out as many times as possible in a minute, you may start with a lot of leg drive, but as form breaks down, it becomes more and more of 2 separate movements. A drive to get the bar moving, then a grind to complete the rep.
For the 5/3/1 approach you could also combine the two movements. Base your training max off of the strict press and complete those reps. As soon as those reps are done, you could hit 3-5 extra Push presses (without re-racking the bar).
I think for overhead stuff, technique (as in getting a good rack position and a strong core/upper back to you can use every bit of leg drive you initiate) will take you farther than just about anything though. The best exercises that have helped me with this are:
Waiter's Walks (I use a yoke but you could easily use DBs or KB's)
Log press with KBs suspended from bands
For odd objects or a keg, I think efficiency is key. For that reason, I attempt to do a hybrid snatch/pres type of movement. I have seen that Kalle Beck video and a LOT of people use that technique. But I do this:
My technique has improved a lot since then (that was 2014 Maryland's Strongest man) But basically, I start way on top of the keg like I am laying an egg.
Then I get my breath and brace.
Then I hinge the keg up to my hips and explode them forward as if I were doing a snatch or Kettlebell Swing. That raises the keg to around my chest level.
I explode my hips hard enough that as the keg rises, I naturally lean backward and bend my knees. That sets me up for the press.
I try to keep the momentum moving and incorporate the leg drive in harmony with the rising keg. The arms start to take over and I lock out my legs while punching my head through.
It is definitely more technical, but it is more efficient and faster which will lead to higher numbers or getting to your next implement sooner.
Here is a more recent video of what i am talking about:
Skip to 1:45