OK. The wobble was why it got hot. If you want to disect it further and post photos, maybe it will help others.
When you install make sure the studs are insulated from any metal dash parts. Some use fiber washers, others go through the circuit board. I don't know your cars dash just mentioning it.
The bulkhead connector - yes. If you can reinstall the terminal so it doesn't move, then maybe its fine. If it can move then the vibrations could cause the screw and crimps to lose grip. A lot of us are dismissive of the wire supports, removing them when they get in the way of repairs and maintenance. Only years later we find out the hard way that the guys who built the car had a reason for putting them there. I for one am a lot more humble these days.
The bench test was a way to reveal resistance if you were reusing the old one.
The concept can be used broadly so its worth reviewing anyway.
Remember that current is electrons moving.
Voltage is level of energy.
When flowing electrons encounter resistance, power gets used and the energy level drops.
On a slightly different subject-
I don't know how the cables are laid out on a '63.
It may be worth looking into using a later style positive cable that has a second wire going directly to the starter relay.
The only thing it might do is provide a more direct connection in a cooler environment. On a slant six this is pretty open and maybe doesn't make a difference.
The other thing to consider is installing a fusible link. The purpose of the link is like a fuse. Its a smaller gage wire with special insulation. The idea is that it can handle occassional, short periods of excessive current ( 40 amps) but if the battery is accidently grounded, it will melt immediately.
It probably would not have helped prevent the overheating failures you saw.
I can't picture this 'webbed' wire you inherited and removed but have to think whatever caused that damage also may have damaged the ammeter stud connection.
The easiest type of link to add will be one with a ring terminal large enough to go on the relay stud.
In the Chrysler Tech booklet we can see a double link used in '65 for the Fury that also protects the horn relay.
Electrical Accessories & Service (Session 207) from the Master Technician's Service Conference
I don't know if anyone repops that or the Valiant/dart version, but if you really like the idea you can buy a 16 gage link and make your own.
Here's two versions with a ring terminal
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