As far as it has been explained to me (and my fallible comprehension) liturgical service IS teaching, but as we see in the scriptures and the liturgy, some things are allegorical, romantic and even historically inaccurate. But this does not negate truth and lesson making.
On the subject of Emperor Constantine, I have heard that he was baptized near death (at the end of his life). Could this be taken as "on his deathbed"?
I can not recall the exact date of his feast day so I'll look it up and get on my menalogion to read the Synaxis and Troparion/Kontakions to see what they say. I'd be willing to bet that we will find it not being stated as to the exact timing of his baptism. This would not be a lie, just something left out.