ask yourself why do the cables look melted?
What causes wires to get hot enough to partially melt the insulation?
Since no battery voltages were given, I can only assume that the new battery has at least 12.50v at rest, and at least 10.5v while cranking. Yes I have seen TONS of "new" batteries die/dead. How long do they sit on the shelf? Who knows? (Used to do dealer work so I saw things that few get to witness)
I have a theory. Without the battery voltage readings I can't say for sure however I'll ASSume that the no load voltage is 12.5v+ and the loaded voltage is 11.25+. Since the choke is not adjusted correctly that tells me that the engine has seen excessive cranking time trying to get it to start. Two things happen. One, the wiring gets hot. Two, the starter dies. The original starters were inadequate for mower usage. The updated Kohler (and kubota) starter is MUCH beefier with a larger shaft assembly and approximately a 40% increase in starting torque/speed. It is a nice upgrade. Or you can be cheap and buy a few dozen chinese starters, and you will have to buy more than one. The Kubota/Kohler starter is the best. Lastly maintenance. The SV840 engine does not have hydraulic valve lifters meaning you will have to adjust the valve clearance sometimes. If left alone (lack of maintenance) the valve clearance will open up which causes the engine compression pressure to rise considerably which puts more load on the starter, wiring, battery, solenoid leading to OE (and chinese replacement) starter failure, wiring problems, solenoid problems, etc.
There's an easy way to diagnose which I've outlined before, search the site for it. These things are super easy to diagnose a no/slow start. I loved working on them at the dealer. 4 years of warranty and the only time we ever saw them at the shop was for warranty repair. Warranty pays a certain amount of labor and on no-start conditions typically 0.5 hour diagnosis and maybe if we were lucky 0.4 hour to replace a safety switch. It took me longer to get the non running mower TO the work bay than it did to diag a no-start condition. We got paid pretty good, or at least I did. The other techs couldn't wrap their brains around a voltage drop test.