The Mazda 6 has minimal grounds. There is a bad area under the air box (see pic). Water gets splashed up from the front tire when going over small puddles. Over time it gets corrodded and burnt. I shortened it and added a piece and ground it up higher where battery hold down mount is. Tranny ground is mainly for automatics. I have the 6 speed Aisin Warner Tranny.
The main fuse relay costs $10 and is worth changing.

 

Make sure they check the charge rate after replacing
the alternator.

 

I am having similiar issues 054 mazda 6 4I had battery light coming on my alternator was faulty charging at around 9volts. I replaced it with a good used oem alternator and i am now chargin at 14.2 at idle. still have a battery light on. I checked batterry tested good i did a load test on battery and alternator and both tested good. With a load alternator drops to 13.7 and does not take long to go back to 14.2 volts. coneections on my alternator are all clean and so is the plug. Everything works perfectly fine lights accessories etc only thing is is i have that dam battery light on all the time. ANy ideas im stumped ??? I have no check engine light just a battery light

 

Make sure ground's are good. You may want to look at ground stock cluster F.uk by wheel well and strip wires back. Also add another ground wire from engine block to battery and to body. Check the stock ground connection as well as battery terminal connections.

other thing is ECU controls voltage, but yours seems to be fine.

Other possibilty is that dash unit has fault in a circuit that needs to be re-soldered.

 

Get a code check on the PCM to double check its happy with the alternator.