I am glad you had such a wonderful trip. I have been to Mt Vernon many times and always enjoy it.
As you learned, pushing a wheelchair can be a bit limiting. On my last trip to Yellowstone without a power wheelchair (in 2014), we borrowed a wheelchair from the visitor's center and my sister pushed me around the boardwalks in the Old Faithful area. That was trying for her and occasionally for me as well.
After that trip, I started the process of getting a power wheelchair. I found that power wheelchairs have changed my life! I now have five!
I sure thought getting a power wheelchair would be an admission of defeat and would dramatically limit my ability to go places. It is just the opposite. Having a power wheelchair has allowed me to conserve energy so that I can do more of the things I really want to do. And, I have found that I can go many places with the power wheelchair.
I have a few threads about my experiences buying and upgrading power wheelchairs. In short, be proactive. Pick the kind of wheelchair that will work the best for you, as opposed to the kind the wheelchair vendor proposes. I have found that front wheel drive wheelchairs are my favorite. While I also enjoy rear wheel drive charis, there are very few practical choices for those in the US. I avoid mid wheel drive wheelchairs, because they can suffer from caster jerk and can become more easily stuck when riding on uneven terrain.
In the US, wheelchairs are classified into several groups. Most private insurance and Medicare will typically only fund up to a group 3 wheelchair (such as a Permobil F3). I find that group 4 wheelchairs (such as a Permobil F5) better suit my needs. Private insurance bought my first group 4 wheelchair, but I have bought 4 others off of craigslist.
I have used a lot of public transportation. There is a regular bus service that stops within a few hundred yards of my house. The buses run every 30 minutes in the summer and every 15 minutes in the winter (I live in a ski area). The buses are free and every bus will accommodate my wheelchairs without problems. They also have a van service that I can call (with 24 hour advance notice) that will pick me up at home and take me anywhere in town I need to go. It has been great.
I have also ridden busses at national parks, such as the Grand Canyon. My wheelchairs fit on those busses as well.
I have taken four flights with my wheelchair. That can be more of an adventure, but was worth it for the trips we took.
We purchased a used wheelchair van shortly after getting the first power wheelchair. I encourage you to get the wheelchair first and then get the van (if you are considering getting a van). Not all wheelchairs fit in all vans.
Wheelchairs have been very liberating for me. I typically put about 1,000 miles per year on my fleet of wheelchairs. It is great fun to do so!
Steve