Here's why it's a bad idea:
Let's say you have a city that produces 20 hammers per turn... and you want to build a unit that costs 60 and a building that costs 120... It would thus cost 3 turns for the unit and 6 units for the building bringing us to a total of 9 turns for both.
Now, we cut the production in half, it's going to be 10 hammers going to each... so it takes 6 turns to train the unit instead of just three... and it takes 12 turns to construct the building assuming that you line something else up after the unit is built... let's say another 60-hammer unit... by the end of 12 turns, you'll have two units and a building... and using the current method, by the end of 12 turns you'll have... two units and a building! The difference is that using the current method you could have two units in the time that it would take to build one by splitting production and you would get the building after exactly the same length of time...
Of course, if you factor in food production for military units like you do for settlers & workers, this might make the system more advantageous for the player, but it'd also slow down the growth rate of cities which might leave you right back where you started...
What I would like to do is have State vs. Private production... it would work something like commerce... each city would produce a base amount of raw production, but then that production would be split between state production and private production. "State" production would be things that the player controls... private production would be things the player has no control over... and it would be adjusted by civics... so under State Property, for example, the player would have total control over a city's production, but under Free Market, the player would have very little... but th player also wouldn't be responsible for building stuff like Banks and Supermarkets... And under Free Religion, for example, the player wouldn't be able to build temples, but the private forces in a city could...
Basically, the loss of control would be the new negatives for the player to certain civics that, for the society, really have no negatives.