Yes, a Britrail pass is good for the Gatwick Express and other trains that serve the airport. However, if you can commit to specific departure dates and times and book well in advance, you'll be better off buying point-to-point tickets. While walk-up fares for long distance trains in the UK are very expensive, advance purchase fares can be very cheap. For example, York-Edinburgh walk-up fares can be over £100, but the cheapest advance fare is only £11.50. Those tickets are non-refundable and non-exchangeable so they won't work if you need flexibility in your travel plans.
Book all your routes at www.nationalexpresseastcoast.com. The site accepts North American credit cards. The discount fares usually are posted about 60-90 days in advance. You'll find the farthest out booking date on the right-hand side of the home page. Book ASAP to have the best chance of getting a discount fare.
For the GEX you can get an up-to-4-for-the-price-of-2 special (no advance purchase necessary).