I missed the message about the result changing with external cooling.
Still, odd. I'd think up and down should behave similarly. Also, even with Ookla's inefficient speedtest code, getting bogged down on a rather modern i5-8265U is a bit surprising. Plus, you said it's the same on the other, more efficient, speedtest sites.
Well, the download seems to stress the CPU a bit more than the upload (on other machines as well).
And, if the fan is not working at all, is not that surprising, since it gest to 90 degrees Celsius even on lighter tasks.
Regarding working without a fan: it's a 15W TDP CPU, and on light loads probably far less. It detects the temperature and probably throttles this way or another (you could inspect how the clockspeeds differ with more/less cooling). I think I recall something about CPUs being able to run even without a heatsink, and that was 10 or more years ago.
Well, that's probably why it has not been fried yet. A 45W+ TDP cpu probably would have been toast (literally) a long time ago.
Anyway, open it up (hopefully not too difficult though it could be), get a suitable fan from China, and replace it. While at it consider checking and replacing the thermal paste too, but that may be messier and possibly unneeded.