Regarding eTap batteries: Incredibly simple to charge. 5 seconds to take off the bike, pop into a charger for an hour, 5 seconds to reinstall. If you were really concerned, you could buy extra batteries and carry them in your pocket. While they last numerous rides, I would imagine the pro team mechanics have lots of extras. Having said that, one of the guys I do a suffer fest group ride with had his eTap rear derailleur battery fail last night. I got the sense that he hadn’t charged it in a long time so he was stuck in his 27 tooth cog not able to access the 30 and 32 tooth for the remainder of the intervals.
Regarding the eTap wireless protocol, it is very reliable in the rear derailleur. But I find it less reliable with the front derailleur. It’s hard to say how often failure occurs (i.e. doesn’t shift exactly when you shift), but I’d guess its about 5-10% of the time. But the key point is that the Shimano wired shifting is always reliable 100% of the time.
btw: I can visualize your teammate’s Di2 battery failure and the challenge of recharging it. It’s pretty challenging doing so even in my garage finding the port; more difficult on the road. A friend of mine had his Di2 battery completely fail so he had to bring his bike to the shop to have it replaced. Not an issue with SRAM.