The client follows the zcash code, currently there is no total supply provided by the RPC calls.
However, you can verify the total the following way.

For the premine of 0.578% we set three blocks #2 #3 #4 at 40,000 each. that gives us 120,000. we mined till block 121, so add 118 blocks of 12.5
that totals 1475 coins, for a total of 121,475 coins as posted on the OP. Why we mined till block 121? we needed to confirm the first few blocks and test TX's both
on T-address and Z-address, then after confirmation of successful TX's we placed a checkpoint at block 120.

Block 122 was mined by suprnova pool and that can be verified by the pool. All blocks after 121 were mined by miners.

Now how to verify that in the client.

You can use the RPC call: getblocksubsidy [block#]:


Now for the total supply part:

We already verified blocks 1 -121 = 121475
Total supply = 12.5 x (current_block_height - 121)  + 121475

So at block height of 5200 we get the total supply as:

Total supply = 12.5 x ( 5200 - 121) + 121475 = 184962.5 ZDASH

All this can be verified as well by checking the code on github.

The fact that Zdash allows storing of coins in private addresses does not change the block subsidy.
We are not in possession of any shards of the key that would create coins from thin air neither. That has been destroyed by the zcash team.

I hope that answers your questions.