I assume the reason the factory manifold downpipe has a braket attached to the bellhousing/gearbox is to isolate NVH and improve longevity of the system.
Without this, the manifolfd and exhaust run to the back of the car, hanging off rubber hangers. The whole exhaust may be twisting as the engine (on rubber engine mounts) goes in a different direction to the body.
Stainless steel welds are also more brittle than mild steel welds. Japanese aftermarket headers often use Goodrich tubing (flexi joint) bolting up to the cat. I think this is to give a degree of tolerance in fitment, and to reduce vibrations to the manifold. The one piece manifolds don’t achieve this. I recall back in the day, the US made Jackson Racing headsers having problems with welds cracking. I have the MX5parts 1.8 Mk1 header. After a couple of years, it cracked on one of the runners, at the cylinder head flange. Luckily, I was able to get it welded, and its been all good since.
A 1.6 I had came with a typical Chinese 4-2-1 header, likely identical to the MX5parts 1.6 item. The car had been mildly lowered, and this header seemed to hang a little lower than stock. Evidently, it had kissed a few speed bumps over the years, and this was enough for one of the welds to open up. For the effort of removing the header, finding a welder, who would need to clean the metal before welding, as it was stainless steel, it was easier to get another identical header for £100, and just replace it.
If fitting a header, I would take the entire exhaust off, and start from the front of the car, rather than try and force a fit where the end of the header is not quite in the same place as the rest of the exhaust. A Mazda dealer will do it by the book, and will make disclaimers because the exhaust is aftermarket.
Easy diy job. With a couple of axle stands down one side, I am able to wiggle under the car, and remove/refit an exhaust. At the moment, none of the nuts will be rusted, so should be easy. The tyre jack makes for a handy third hand at certain stages.
Disconnect the cat pipe from both ends (or loosen the bolts). Fit the front of the cat to the header (check if the garage fitted a new gasket or not. I’d fit a new gasket, making sure the flanges were perfectly clean.
The Cobalt system as I recall comes in two bolt together sections. Undo these sections. The mid section will now hang from the hangers (post-1995, the front hanger was deleted). I think it will be easier to unhook from the hanger (soapy water helps). Connect to the cat pipe, like before, with fresh gasket. Hook back up to the hangers.
Now the back box. This is where it is handy to have something to rest it on, once you take it off the hangers. Reassemble to the mid-pipe. Now lift it up and hook back up the the hangers. Here, I find I need to do the most tweeks to the rubber, to get the tail pipe to sit centrally and not too low/high.
Places to check for rattling; the mid silencer might be a bit close to the PPF. The rear silencer or tailpipe might hit the rear tiedown.