Old MacBooks no longer boot Snow Leopard
That photo says the kext (driver) for the Atheros airport card is what's causing the kernel panic.

There's a very simple way to disable this kernel extension, although it's terribly hacky: go into /System/Library/Extensions/, and rename IO80211Family.kext to IO80211Family.kext.bak.

Afterwards, go into the Terminal and run sudo kextcache -system-caches to rebuild the kernel cache.

On newer systems, this would require disabling SIP and/or remounting the root partition, but this is Snow Leopard.



Since you don't need the wifi, the above may be enough for you, but I'm at a loss as to the root cause of the problem—why would this kernel extension suddenly begin panicking? Safe mode, recovery mode, and single user mode likely don't load this extension, so the fact that those work makes sense, but how could the problem have suddenly appeared on two different machines?

The only other suspicious thing is that the computer kenel panics right after loading the lego mindstorm kext (com.ni.Fantom.nxtfwdl). That kext isn't listed in the backtrace at all, so it shouldn't be the culprit, but... if you upgraded the mindstorm software recently, you might want to try downgrading it. Or, you could try removing the mindstorm software altogether, but it sounds like that would render these machines useless to you.