I definitely wouldn't describe it as baby sitting. For me it is every 1000km or so for my SRAM rear derailleur, or about 1500km for my Shimano di2.
So it is around once a month if I am riding that bike.
In terms of reliability, it has been significantly better than cable, never had a a failure with electronic shifting. Have had campagnolo eat my shifter cable twice, leaving me around 80km from home with no rear shifting. And a huge pain in the ass to get all the cable out.
Edit - oh and batteries for the shifters, last year's.
The only non rechargeable ones are the coin batteries in the shifters, and after over 2 years and more than 30,000km on that bike, I still don't need to think about changing them.
The rechargeable batteries are simple, much simpler than things like my lights or my headphones, watch, phone, every other item that you have to charge constantly, because they last so long. My Garmin also tells me when it is getting low on charge(and that's still enough to comfortably ride for 200km+)
Couple that with how much better and consistent the shifting is, you never have to touch it again, no cable wear or changing the cables, never have to deal with bad shifting, I will never go back to mechanical shifting.