You can try buffing out the scratches with a rotary buffing tool and the appropriate buffing compound. Not unlike how one would wet sand, buff, and polish out scratches in automotive paint. There are also specific products meant for repairing scratched porcelain sinks and etc, but I’ve not used them. It’ll be substantial work, results dependent on skill, “like new” outcome not guaranteed.

That is very frustrating. My son decided to help me by cleaning my polished copper pots with a green scrubby. I set him up with a buffing wheel and buffing compound to repair the damage. It took him eight hours and the scratching is not fully gone.

How expensive is it to replace the scratched part?

That appears to be quite badly scratched. Just by looking at it, I wonder just how many of those scratches will actually buff out. You might be wise to get the replacement of the porcelain top. Some of those scratches look like they go all the way through.

Try to blend it for consistency with more Brillo by hand. You can't make it much worse.

Throw away the Brillo pads and get a new top. You're not polishing those scratches away.

Paint over porcelain on a stove top isn't a long term solution...not even a short term one. Paint is not going to hold up to the high heat nor the cleaning necessary to remove heat polymerized grease/ varnish.

There are high temp paints (exhaust manifolds etc) but unsure how they’d hold up to cleaning.

I still think removing the piece and going at it with rotary buffer and buffing compound is worth a try. If results not good enough, replace it.