I'm always interested in ideas that might open up the higher levels of the promotion tree a little more often, instead of extremely rarely or never. However I don't think this one would work as it would take away the crucial early decision about whether to build the barracks before the first few units or after them, if they could just come back for basic training later anyway.
But I've always had a soft spot for the idea of another "plain" XP building in addition to the barracks and stable. Something maybe around gunpowder like a "shooting school" which gave another 4 XP, would just prise open the promo tree a touch more and make things more interesting IMO.
I realize this would affect game balance by relatively weakening the other sources of XP and promotions (civics, agg/pro, UUs, GGs, wonders, etc.) but not in an unsurmountable way. I won't hijack this thread any further by speculating along those lines though.