I’ve used giffgaff for years, and would recommend it. The £7.50 goodybag is what I use as my standard monthly ‘plan’, and I never get through all my data, so I imagine it will be enough for you for a visit. One nice thing is that if you find you do need to phone the US, giffgaff’s rates are low, only 2p/min.
City-Go seems expensive, since you’re essentially paying for things that are free (SIM & Oyster cards). If you want to get Oyster cards, it’s simple to do at any Tube station from one of the machines. There is a deposit (£5?), but it’s also simple to return the card at the end of your stay. The larger machines at stations have this facility and will refund you in cash for the deposit and any remaining balance.
Saying that, I’ve been using my Oyster card a lot less on trips to London as, like others have said, a contactless debit card works just as well. Since my bank card is contactless, it’s one less thing to remember or top-up. Similarly, Apple Pay works well. As @aforkosh says, make sure to use the same card in Apple Pay each time to take advantage of daily/weekly capping. So if you all have Apple Pay and you get a good exchange rate on the card you use it with, then just use that. However, if the exchange rate charged by your US bank for foreign transactions isn’t good, you’ll be better off converting money and loading it onto an Oyster card when you arrive.
TfL provides an overview of using non-UK contactless cards. From that it sounds like there can be issues with using some US cards directly at the gates, even though they can be used to top-up Oyster cards at the machine. As long as you have a bit of time (10-15 minutes), you can try this out when you arrive. If your preference is Apple Pay or your bank/credit card, just try it and see if it works. If not, you can easily go to a machine and buy an Oyster card. Whatever you do, don’t buy individual tickets from the machine, as they cost a lot more than using Oyster or contactless. It’s all a lot simpler than I’m making it sound, and in my experience the staff at stations are very helpful (they ‘mill about’ so you can approach one and ask for help if needed).
Regarding registering your card with TfL, this is not necessary, and I’m not confident that registering a card will guarantee that it works. For instance, you could probably register a non-contactless card, but you won’t be able to use it at the gates. The good news is that you can register your card at any time, even after you’ve started using it, and you should still get historical journey data (I believe). With Oyster cards, you can also get a list of historic journeys at the machines at stations (even if the card isn’t registered), I’ve never tried this with a bank card, so not sure if it works with them.