There are cheap braided lines that do impart a bad taste to the water. There are better braided lines that don't. You have noted that the water from the tap is fine, and the water from the hose is bad, so you have isolated the problem to the hose. Just get a better hose, and read the reviews before you buy. I've used the Eastman brand from Lowe's without a problem. YMMV

I don’t like copper in this situation, too easy to crimp or bind. I usually use the grey plastic water lines from the box stores. I assume they are some kind of pex

I would say the bad taste is coming from bacterial growth, and the braided line has plastic inside so who knows what stuff the china company used in side to make the braided tubing.

Anyway Just like well water with bad tasting and smelling water the remedy is to shock the well water and suction line (plastic) with with chlorine. If you can pump some chlorine into the system and let it sit awhile and then flush it clean it may solve your problem. Once bacterial growth attaches to the inside tubing it's hard to just rinse it away.

If you shock the entire line from the tap to the frig outlet and the bad taste goes away it's a sign it was bacteria. If it's still funky I'd say the inside plastic is decomposing and releasing a nasty chemical. Your plumber is dumb wrong, Copper is antibacterial and few if any living bacteria can live on it's surface. Google "is copper antibacterial?" and see I'm not blowing smoke up Uranus.

Skeptical of copper, unless you have access to disconnect/reconnect before and after pulling the fridge out, or leave giant loops for strain relief. Just get a better braided line, and monitor for leaks after installing.

Have you checked if the water coming into the line has a foul taste?

You do understand that the braiding is not a direct factor? It's a separate sheath on the exterior that encases the plastic interior tubing.

Try a different brand of connecting line as others have advised. And another if necessary until you find one that's satisfactory.

Yes, I mentioned that the water coming from the tap/sink where the braided line is sourced from is fine.

Several owners of this GE Water line reported bad tasting water in their comments. An owner of this Everbilt water line that water from his previous line tasted bad but that switching to the Everbilt product solved the problem:

" Resolved my issue of bad tasting water on my new LG refrigerator. The braided water line that is recommended and installed upon delivery has a rubber based lining and when the water sits in the line for any time. This water line has a PEX lining and eliminates any bad tasting water (plastic taste and hose taste) issues you may have. Happy with my purchase."