Older thread and it doesn't look like the OP has been back in a couple months, but I'll throw in my 2 cents.
I had a look at the vid, but the only part that matters is the first hook cycle. When the second happens, you've already got a mess going on. So it's not going to get any better, it's only going to get worse. Everything actually looks fairly normal until the hook has to pass off the thread.
Off brand, but it appears to be a singer clone. Looks likes someone tried to put the Singer "sphinx" graphics on it to make it look "purty" and just covered over the "ER" when they ran out of room.
That's an oscillating hook, so the hook is supposed to run a little loose in the race. The center "pin" wobbling isn't a big deal, it's part of the hook assembly and moves with it. It's going to do that because the hook needs to fit loosely in the race in order to not jam and to allow the thread to pass through where the driver contacts the hook. Some drivers have a spring on them to take up the clearance and some, like this machine, do not. Neither has any real advantage over the other, except that a spring loaded driver is usually installed by the manufacturer to quiet down a noisy machine if it rattles a lot without one. If they can get away with not using one (IE: save $$), they will.
It's hard to tell what is actually happening without a bobbin case in there. Without one, it will catch like that every time, so the first hook pass is about the only part that shows what it is doing.
But if it catches the center pin with the bobbin case installed, I would say the machine is out of adjustment or the bobbin case is not the right one for that machine. Or possibly, the hook is not the right one. A bobbin case too short or a hook with too long a center pin could be the problem. That's the thing about old machines and new reproduction parts; they're not always manufactured to the specs needed to make the machine run right. they can even look the same, yet be off just enough to cause a problem.
If it's the wrong bobbin case, it might not be covering the pin completely (as it should when installed) and allowing the thread to catch the pin. Same same if the pin is too long.
Wrong bobbin case is an easy fix, you just need the right case. Same if the hook is not right for the machine.
But "easy" is a relative term here. It's easy to fix, but being a copy, finding one that fits your machine might be a challenge. Sometimes you can use parts from the original machine that they copied and they work, but lots of times they don't. A copy is just that; a copy. It's might be "mostly" the same, but not usually not identical. Some times you get lucky, some times you don't....
Out of timing could be another issue. to me, it looks as if your hook driver is not rotating around as far as it should. You may also have an issue with the thread take up arm (little arm that goes up and down as you sew and holds the thread). The way an oscillating hook machine works is that it keeps the thread taught as it enters the fabric. The thread take up arm and tension discs enable this. When the needle starts to come back out of the fabric, the thread take up arm goes "slack" and the thread forms a loop down in the bobbin well from the friction of the needle moving up through the fabric. Then the hook comes around, passes through the needle scarf and grabs that loop. The driver moves the hook assembly to the bottom, where the top thread is supposed to slip off the hook. At the same time, the thread take up arm now starts to pull the thread up and out of the bobbin well. this pulls it around the bobbin case and captures the bobbin thread. When the whole shebang gets to the fabric at the needle plate, the take up arm pulls the lock stitch tight. It can do this because the tension discs now prevent the take up arm from pulling thread off the spool, so the arm ends up pulling the top thread tight in the material and setting the lock stitch. Then, the feed dogs come up and move the fabric back and the whole cycle starts all over again.
The hook driver has to rotate far enough to "pass off" the thread so the take up arm can pull it out of the bobbin well. If either the hook driver isn't rotating far enough or the take up arm is not timed correctly, you get a big mess down in the bobbin well. My Singer 331K105 had a similar problem, but it wasn't handing off the thread consistently. It was just a small adjustment to the hook driver, but it messes everything else up in the process. this required me to start form square one and realign the entire machine.
It's a bit hard to tell from the vid, but your hook doesn't look like it's timed right. It might just be the video angle though, as the hook does seem to pick up the thread loop fine (at least on the first pass).
Hook/needle timing is important, but it's not the only aspect of the machine that can come "out of time".This is where a good service manual is helpful as they often lay down what and how to adjust, but it usually also gives you the SEQUENCE they must be adjusted in in order to get the timing right.
