Sure, I'm not saying it's a good idea, just that I don't see any inherent problems with it - given that fly-by-wire systems have been used for decades I would consider them well understood.
And would that affect its capacity to glide?
Not in this case, since the instability is caused by the lift generated by the engines.
IIRC, the instability is caused by the engine nacelles (the metal shroud around the turbofan), and therefore is present whether the engines are active or not.