satoshis reason and math for 21m is not what you think. nor does satoshis decisions or math have anything to do with your math
even cores current code for checking the 21m is bad and insecure.
it actually starts at a satoshi amount of 5,000,000,000 per block (yep check any and all block reward value in 2009 they all start in binary amount of 5,000,000,000 units not 50 units)
which after every 210k blocks the reward divides by 2
in fact and in short its actually a binary amount of
100101010000001011111001000000000
represented in hex of
12A05F200
which then displays in numeric of
5,000,000,000
which every 210k blocks the binary reward amount loses a bit
100101010000001011111001000000000 (5,000,000,000)
becomes
10010101000000101111100100000000 (2,500,000,000)
becomes
1001010100000010111110010000000 (1,250,000,000)
and so on
which if you add up all rewards and average out the timeline of the average block (secured by difficulty to try to keep them at 2016 block per fortnight(~10min/block)) then means after 32 halvings(because there was 32 bits in the first reward binary amount) every ~ 4 years, mean it totals 21m(rounding error) btc or more specific 2099999997690000sat at around 2140
What you are talking about is a specific implementation of approximating the logistic function. To implement it, you first need to conduct mathematical modeling and define the function you will be approximating. The choice of 21 million is not random. Satoshi could have chosen any parameter W.
The generalised logistic function or Richards' curve (1959) was developed for growth modelling (extension of logistic). The model has been modified by Sugden (1981) and Satoshi (2008)
Wt[t]=W(1-(1-m)Exp[-k(t-T)/(m^(m/(1 - m)))])^(1/(1- m))
W=21000000
k=0.002625
m=0.3909
T=165.6