Best product to get my kitchen sink back to new looking?

They come in all flavors and colors. None work. Experience.
Replacing it with new is the only way. No shortcuts.

Nothing is going to get your black sink back to its original shine. If the dullness is due to hard water build up, try a soft cleaner with bleach. I personally like no scratch comet with bleach. I rub it in and let it sit for a little bit, then wash it off and wipe with a soft cloth. Takes the hard water dullness right off of my sinks. You could also try Mr Clean Magic Eraser.

Using comet.
After you wipe clean, use a window cleaner to boost up any shine.

After cleaning use some Orange Glo Wood polish.!! A little goes a long way!! Try it it works!!!

I use a 50/50 mixture of vinegar and water to polish it up. I wipe it with a paper towel until it’s completely dry and shiny. All my appliances and sink are black.

KABOOM in the purple bottle (spray)

use a brush with bristles to scrub. Works like a charm