You probably want to be able to use digital modes. Your best bet to get consistently across the country will be via digital modes, as they can add something like 20-30 dB effectively to your signal. Olivia in particular might be a good choice. This is achieved because digital modes are more sensitive, able to read signals below the noise floor.
If you want to be able to reach your "neighborhood" consistently, and occasionally around the world, then your best bet is probably 40 m. In fact, 40m is a highly recommended band to start out on. It works very reliably up to about 700 miles during the day, a bit further at night, and if you get it high enough, can work really well at night. Plus it's less crowded and generally more friendly than 20m.
20m is the best band overall to work around the world, but you often will have difficulties with things that are closer.
If you really want voice to anywhere in the world, you'll want a huge yagi antenna on a large tower, transmitting at the full legal power in a quiet RF area. Other than that, you will have some compromise.
My set up is a G5RV in my attic, which I can talk routinely to Europe on 20m, most of the East part of North America on 40m, and occasionally further on 10 and 15m. I live in Virginia (FM19), to give you an idea.
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Based on the parameters given I would say 20M is the best band to aim for. There's usually a lot of activity, it tends to give reliable propagation, your antenna can be relatively small and there's a little bit of everything happening on it (CW, voice, digital and SSTV). From south-central Canada I've worked as far south as Cuba and as far east as Germany on a horizontal dipole hanging about 3 meters (10 feet) off the ground and most of North America with a sloper hanging off a tree at the edge of a field.
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20m is the "go-to" band for long haul comms without a huge antenna, but it is by no means the only one. I've had 5000mi contacts on 20m, 17m, and 10m (using only 100W transmit power). With more space for antennas, 80m and 40m have even more potential for long distance.
"unconditional access" is the hard one. There is no one band that will do that for you all the time. It all depends on the atmospheric conditions. For instance, 10m is only good during the daytime, and 20m is best during the daytime, and I wouldn't expect long haul comms out of it at night. To pick 3 bands, 10,20 and 40m would work if you learned how propagation works. You can also build (from speaker wire) an antenna that will work pretty well on all three of those bands