Just to increase my knowledge in general, I did some reading/searching and looking at your links. So, someone built one with a 9V battery, a variable resistor and an analog meter.
Some things that came out:
9 V battery is somewhat underpowered
2 mA max
negative potential to skin
Discussion about electrodes (sponge and real electrodes) - No one mentioned Titanium
Time duration? I (guessing order of seconds)
An analog current meter (e.g. 50 uA) runs $100 USD You can MAKE them read whatever (e.g. 2 mA) with resistors)
Parts do add up and so does postage.
A GFCI (
https://www.ul.com/wp-content/uploads/2014/04/ul_GroundFaultProtectiveDevices.pdf ) trips a 4-6 mA. So, that amount of current across the heart muscle can be deadly.
We USUALLY consider 24 VDC and power limited to 100 W to be safe, but poking a needle in the skin and using an intentional electrode patch well...
So, the needles, the needle holder really need to be commercial. The BNC connector seems to be common.
There is a current regulator chip available
https://www.linear.com/product/LT3092 but a resistor is obviously safer.
This
https://www.linear.com/product/LTC6993-1 chip is probably better than a 555 timer IC, but it does require a 5 V supply.
So, I won't design one because safety is a big issue here.
You would want some sort of LED indicator that the needle is powered. Since current set in the case I described is somewhat arbitrary you may need a push button way to keep it on, until the current can be adjusted.
The same would be true with the time. Adjustment could be fixed or variable with the knob position indicating the time. It could use a turns counting dial.
I would not go higher than a 24 VDC source. Charge/Use modes again adds more cost. The 9V battery battery by itself would require constant adjustment.
A low battery indicator would be a nice addition.
I'd make a comment that it would seem you would like the large patient electrode to be near the area of hair removal. i.e. Not use your left ankle to remove right arm hair and not to intentionally span the heart muscle.
Self doing it means you no longer have the operator essentially being independent and can remove the needle.
Researching RF, I was 100 V p-p at 13.56 MHz. 13.56 is an industrial freq where you don't need a license. Again, all the variables are missing. Max current, Max power, max time etc.
Yep, batteries are definitely safer by a long shot and so is a a variable resistor 9V battery and an analog ammeter. And so is having a second person doing the procedure.
This
https://forum.allaboutcircuits.com/threads/oscilliscope-purchase-help.123313/#post-991452 guy has an interesting view of safety.