Far to many political leaders had a incentive to increase their voter base by extending the franchise. The growing urban populations who more often rented were a attractive addition to the polling registrar.
If in fact the property requirement remained firm, then post Civil War the freedmen would be far less a threat to the small landowning rural southern upland farmers, thus reducing the post war political & social tensions and the 'Terror' of the latter 1860s.
It is also possible the working classes would have a greater incentive to own a small house or tiny bit of acreage so as to gain the vote and patronage of the local politicians.