Battle Familiar

Can a Medium size familiar such as a zombie that has Battle Familiar cast on it with the "Flyer Form" act as a flying mount if my character is Small sized?

The normal stat block for Zombie (found in appendix B of the PHB) is a CR 1/4.  This means the CR is too high.  The stat block for the Battle Familiar changes the CR to "None" which was weird enough to make wonder. 

I suspect the answer is "No" but I think it would be funny to have a flying zombie haul me around a battlefield.  I guess even worse is that if I use a Deer as a familiar I can no longer ride it if I cast Battle Familiar on it as it would have it's stat box changed as well.

I don't have the details of Battle Familiar, but a Warlock with Investment of the Chain Master can have a Skeleton and infuse it with a Fly Speed, so technically is possible.

Now... there is an additional requirement: "You can control a mount only if it has been trained to accept a rider." (EDIT: I'd allow it as DM)

The thread Wee Skirmisher: RAW or RAI? was about it.

FYI it is important to distinguish using a creature as a "mount" - which generally requires them to be approximately horse-shaped and specific riding equipment for you to sit on/in to be appropriately mounted on them, vs a creature grappling you and carrying you around - which any creature can do as long as it has some way of holding onto you.