Toyota Repair Problems? Ask a Mechanic for Answers ASAP
When the engine is running what is the voltage at the battery? What is the voltage of the large cable at the alt? Honestly the 13.7 on the old alt is normal and consider good not sure why someone told you it was not. The voltage is not the issue it is current flow ie AMPS.
The large cable is the main output cable that goes directly to the battery through a large fuse. Is there any voltage to that large cable with the 10 mm nut on it? Check that with the engine off.
Unfortunately I have bad news for you. You have blown the large ALT fuse which is part of a fusible link strip and is incorporated into the main junction/fuse box under the hood. There are no written directions on how to remove that fuse. You have to disassemble the fuse block and remove that fusible link strip which is bolted to the bus bar in order to resolve this problem. That only happens by inadvertently shorting out the large wire from the alternator. Click this link for some illustrations on how it comes apart this is not the exact same fuse block but is similar it'll give you an idea of what you're dealing with. This is not a job for someone who does not have a lot of patience and good at assembling and disassembling things most shops will charge you about two hours in labor to replace this the fuse itself is only about $40.
You are welcome.