Most of the swivel seats use a support leg instead of the tether, so it doesn't get in the way. Rather than holding the top of the seat back, it holds the bottom of the seat up and prevents it from tipping. What they are designed to avoid is forward rotation, the seat is affixed by the two isofix points which are at the base of the seat, so in a crash the child's weight is mostly above these and the whole seat will tilt forwards, which is dangerous as it moves the child forward too and they can hit their head on the front seat, which is usually fatal for children at crash speeds. With isofix seats, you can have either the leg or the tether, but you must have one or the other, and where they exist they need to be used.
Top tether works well with fixed, forward facing seats. It doesn't work well with spin seats and it doesn't work that well for rear facing in general. Although this seat is designed that you can swivel it with the tether attached in rear facing mode. Top tether is cheaper to produce than a support leg, so you see it on the cheaper brands usually. They try to cram all the keywords into the same product - swivel, isofix, all stages, side impact protection, extended rear facing - and they rarely deliver.
I have no idea what those metal parts are for but given it's a budget model I would assume that they aren't in there for no reason, and would be reluctant to use it without them in case they do some kind of securing the straps in place or something. It might be that they seem to work fine in day to day adjustment and use but it could be in a crash situation that it matters.
If you wanted to change it for something else, Graco Turn2Me is probably the best deal on a spin seat at the moment, £135 on amazon and you have the support leg. It can also rear face longer than the Cozy n Safe, which is only rear facing up to 87cm.
https://www.amazon.co.uk/dp/B08YB5G5ZW?m=A3P5ROKL5A1OLE