I have a brand new Aulos 511 tenor recorder and can achieve perfect overblown octaves without changing my fingering, just adjusting the position of my tongue and directing the stream of air. Why are there different fingerings for notes such as E5 compared to E4?
E5 is usually fingered with the thumbhole half opened because it speaks at lower air pressure than if you use the fingering for E4, with the thumbhole closed. The ease of playing E5 with the thumbhole closed also varies from instrument to instrument, and is perhaps especially easy on your Aulos 511. In any case, the notes above E become increasingly difficult to play without half holing the thumb, so there's no real advantage to not doing it starting at E, except perhaps for some special effects.
First of all, let me point out that overblowing without using an octave hole or octave key Can be achieved with any wind instrument. It was actually a practice technique my clarinet teacher suggested. However, in general use of the octave key to stifle the fundamental allows you to produce the overtone (octave, 9th, whatever) with less effort and much sweeter tone.