I can't see any immediate/easy fire risk.
Hint: One of them is black, the other red. They go directly to the battery.
Also these things get chewed on, thrown around, and who knows what else.
But they would have to break through the plastic case, break the insulation of both wires, then the wires would need to contact each other. Even then, I'm not sure such a modest current (maybe 0.7 Amps), could readily cause a fire.
Or just a strand of a stripped end shorting at the PCB end. These wires will be crazy thin to start with.
On reflection, I agree with you. That is a very good point. That sounds very plausible.
Now, is that 0.7 Amp back of envelope, ball-park measurement, enough to potentially cause a fire?
Looking into, the minimum current needed to cause an electrical fire, finds the following, professional/academic looking, interesting (free/open) paper, on the subject:
https://www.mdpi.com/2571-6255/5/6/201Which seems to mention (but it refers to another person or source of information), that as little as 4.5 Volts (coincidentally the nominal battery voltage, of this threads situation, 3 x 1.5 Volt AG13 batteries = 4.5 Volts), and 300 milliamps (which seems plausible, since you already measured a short-circuit current of 0.7 Amps), may be able to cause sparking (maybe even arcing), and hence be an ignition/fire risk.
Also, on reflection, I've realized that my assumption, that the wires were pure copper, plastic insulated ones. Could be wrong. The Chinese (other countries could have been just as bad), may have cheaped out on these wires, possibly being a different metal, or copper coated metal of some sort.
Hence the wires, could, unlike copper, have higher resistance, be more susceptible to breaking (brittle, hence sparking/arcing etc), possibly even being flammable (aluminum comes to mind).
So, I now think, that it has become more plausible, that those 3 x AG13's (or LR44's, if that is what they were replaced with), may be able to be the cause of that fire.
Practical demonstration. This video, less than 2 minutes long (if inpatient, at around 38 seconds into the video, it is shown), seems to show a low voltage and current, set fire to the ends of some thin wires. I'm not sure of what PP3 battery type they were using, or how much more current, it can make, as a result.