Here is an excerpt from The Complete Book of Thunderbirds 40th Anniversary Edition about it:
Early in August, Barry Gray recorded 'Flying High', a song performed
by Garry Miller for Thunderbirds' end title sequence, but he was
unhappy with the session and re-recorded it twice in early September
. *'Flying High' was to have followed the tradition set by
Fireball XL5 and Stingray where each episode ended with a song that would act as an alternative theme tune to the one used in the
series' opening titles*. Gerry Anderson ultimately decided that having
a song (and, in particular, this song) playing over the end titles no
longer suited the format of the series, so 'Flying High' was scrapped
and replaced by the 'Thunderbirds March' just a fornight before the
first episode was transmitted.
So, as you can see, yes 'Flying High' was originally intended to be used as the theme song for the end of Thunderbirds, supported also by the fact that Anderson's past series' had also used this format of having a vocal song during the end credits.
As mentioned in the comment for that video in the question, that is the original song, however I cannot find any videos of the original end titles with this song in the background; only fan-made versions. It is quite possible, considering the above quote, that these end-titles were never actually made with 'Flying High' in the background!