I read some articles, and 16 pulses are not 100% requirement. as well as precise timings.
So the pulses at 32 kHz is actually the way IR reciever works, it has bandpass filter inside and it is tuned somewhere around 32k. For example in TVs they sometimes have frequencies close to 38 kHz if I remember correctly..
And once you pulse your ~32kHz, the reciever will output a steady 5v (or maybe 3.3v). It doesn't care if there are 16 pulses or slightly less/more.. there is no counter or anything.
The camera itself has some timing tolerances. So you can use any frequency close to 32kHz and send any amount of pulses for ~500us, then wait ~5,8ms and send pulses again.
If no counter and precise timings required, I can make the whole circuit using just simple analog components

Why? just because it can be done

I like microcontrollers but here it looks a bit like overkill.. it is just more interesting to do it analog way sometimes.
Here
https://www.doc-diy.net/photo/rc-1_hacked/ in the comments someone made his version using just 4093 NAND chip, but I don't really like two switches and 6v batteries. So now I have to design my own version from components that I have around.