Dead battery is below 10 volts. In this case, voltage is ok, 12,6 but when is under load there is not enough current. Problem is in connection strips between cells.
I had a similar problem in my car, premium brand battery, just one month after warranty, clicking sound from starter. Jump start - ok. Charging ok, 14,3 v. Starting: trrrrrrr....tk.tktk.
Whitout load, 12,7 v. So, cells are ok, there are 6, good single cell is 2.1 v. Battery is sealed, only plugs for electrolyite.
DON'T TRY THIS AT HOME, THIS IS NOT "HOW TO"...
Batteries are not meant to be serviced. And they are full of acid. But... Hehehe..
Soldering iron, cut holes in top cover above connecting strips, they were just pressed together. Probably bad spot welding. Strips are made of lead, solder the connections, fill cut holes with hot glue, battery works, no leaks. I sold the car two years later, with this battery. It wos ugliy, but it has plastic cover over so... Somebody, someday wil wonder: wtf is this?
AGAIN, DONT TRY THIS. SIGNLE SPARK AND BATTERY CAN EXPLODE.
I use soldering iron and hot glue pistol. They don't spark. Other, grinders saws... No, dont do it.