While hunting in a store, I found a F10T8 tube which was never used and was brand new. But, when I tried it with a 10w electronic ballast, it didn't turn on, not even a flash. Even as I got to my 36w electronic ballast, it still didn't work. But, it works flawlessly on preheat ballasts, but only if I manually preheat it, and blinks endlessly on starters. Why is this so? The lamp does not mention any specific ballast type, and is labelled as a common use fluorescent tube.

How long is it?

What ballast were you using for your experiments? Some cheepeese simple one without any protection or some well protected one?
Because it is possible the tube has bad emission, so the ballast shuts down before you notice anything.
Plus if the ballast is designed for the "standard European" 10W lamp, but you are using the short (30cm) cheepeese tube, it won't warm up properly, as the longer "European" is a 0.17A design, but the short one is something around 0.27A or 0.3A (I do not know exactly), so preheating it on the 0.17A balast would require way longer preheat times (and higher preheating duty cycle for unsuccessful stsrts) than the normal starters are designed for (they count on the preheat current to be 1.6..2x the nominal current, so about 0.25A for a 0.17A lamp, but the short 0.27A lamp will need about 0.45A preheat or so).

The 33cm is the short type.
The question is, whether you ballast is designed for these 33cm, or for the 0.17A 47cm version.
The confusing thing is, both versions are T8 and both are 10W rated. But they differ in rated current and arc voltage.
The 47cm is listed by "tradditional" European brands (Osram, Philips,...), but the 33cm on I have seen only some no-name cheepeese origin.

However I do not have any indication that either of them should have any problem with either ballast type, of course assuming the ballast output matches the electrical rating.
In fact the 33cm cheepeese use to came virtually only with the simple "CFL like" cheepeese electronic ballast.
So therefore my guess was, you are trying some higher quality protected ballast but for the other type and so it keeps tripping the protection so refusing to light.

But then, why do other F10T8s, with the same size, work on that ballast without any problem? Also, when I used a higher rated ballast, it should have turned on like my EOL F10T8s.