In my experience, (Tooth bar vs none) :: (Dirt vs none).
Definitely a good idea to get one.
I've got a piranha tooth bar; it works great for dirt that's reasonably compacted but not actually hard. I think the teeth you got are going to be fantastic as it'll probably be able to dig into even harder stuff, possibly even point the teeth mostly down, push down on the bucket, then roll the bucket back and you'll get a nice chunk of hard dirt out (don't drive forward with the teeth straight down though!).
A nice thing about the PTB I've found is that you can push it into a smaller tree and uproot them, and also I've found it super useful in pulling brambles and jungles of small shrubs without completely destroying the soil by similarly catching stems with the ptb and shoving; the sharpness of the bar is such that you can shave hard stuff, but there's still a lot of bar hitting hard so if the dirt is really hard it won't sink in like the teeth will.
Can't advise ramming the bucket in at all. I understand the sentiment but that shock isn't going to be good in the long run. Front end loaders on little tractors are not built to push hard, they're not excavators or dozers and the design case is for loose material. YMMV and it's your tractor but don't be surprised if you crack or bend something by ramming.
Going for the corner of the bucket - slow and occasionally yes, and make sure you trade off which corner often; note that you're deliberately setting up uneven forces on the loader and something's going to tweak in the long term.