To me it just seems so inefficient production-wise [...] Early game production is vital
Avoid farming plains tiles when running Agrarian. If you only farm grasslands and flood plains then there is no

penalty so there's no production inefficiency.
The way I see it, agrarian without aristocracy is much more effecient for production. Farms produce more

which allows you to work more

tiles like mines or lumbermills. Aristocracy actually slows down settler and worker training, especially compared to God-King.
Agrarian is't any more efficient for production; it has the -1

penalty on farms, not Aristocracy. If you have to farm a lot of plains tiles then you may be better off not using Agrarian, especially if you are using God King.
Of course, without Aristocracy you'd have faster population growth and better worker/settler production. God-King/Agrarian is great for expansion, as long as you aren't forced to farm plains tiles. The price is paid in

, however. With Aristograrian you'd expand slightly more slowly but have a much stronger economy.
There's no reason why you can't benefit from both of those aspects, however. Aristocracy isn't available until
Code of Laws. Although you can get it without getting
Mysticism, I recommend the detour. You can adopt G-K/Agr early to fuel your race to fill up the available space, and then switch to Aristograrian to stabilize your economy.
In your example you're not taking into account the higher

production from farms under a non-aristocratic. Which ties into my complaint that it cripples

. Running specialists eats up even more

. And again if you don''t run aristocracy, your farms produce more

so it's easier to run specialists.
In the long term other approaches will achieve superior extremes. Cottages will eventually mature to produce more

. Agrarian alone will produce more

, which will support more specialists, which will produce more great people. Aristograrian, however, is a more balanced approach. It requires no time to mature into its potential. It produces more food than the cottage approach, and more commerce than the specialist approach.
The biggest draw for Aristograrian, I believe, is Financial leaders. The trait is popular, and with good reason. It makes Aristocracy even more appealing, because of the additional

. If you happen to be playing the Calabim then the draw is even stronger, because God King is unavailable and City States can only compete with Financial-Aristocracy in a truly massive empire - which you are unlikely to have that early in the game.
I use it rarely, [...] And I only use it short-term [...]
It seems alot of players swear by aristo-agrarian and I don't understand why
I hardly ever use it myself, unless I'm playing a Financial leader. Sometimes I use Aristocracy with an elven civ, but never with Agrarian because of the

penalty (every tile is making at least 1

due to forestation and the +1

from AF replaces the +1

from Agrarian). I know why Aristograrian is strong, so I understand why it is popular, but it doesn't really fit my style well. I prefer to focus on

output for my cities, and often will run God King/Agrarian early on switching to Republic/Something later in the game, where "Something" depends on my civ, religion, and whether I'm aiming for an economy based on 0%

or 100%

. I will have one city as my economic center (the holy city of the religion I found if 0%

, my research hub if 100%

) and it will have cottages. I generally don't build them anywhere else because I want the

or

I can get from other improvements more than I want the

I'd get from a cottage.