Specifically Gates and Allen were given timesharing access to a DEC-10 at school - which was an incredibly rare privilege in the early 1970s.

Allen had already written an 8008 emulator on the same model of mainframe as the one at Harvard. And that made it possible to update the emulator while at Harvard and then use it as the basis of their version of BASIC.

Any programmer at the time could have written in 8080 emulator. They didn't have to be a kid in school. The DEC-10 was a popular computer. MIT had computers available to students, too. Gates didn't invent BASIC - it was developed by others for use by students.

Edit: There were many thousands of computers at the time, and programmers to run them. The 8080 instruction set is trivial:

http://pastraiser.com/cpu/i8080/i8080_opcodes.html

and any semi-competent programmer can write an emulator for it in a few hours. The magic ingredient Gates and Allen had was Recognizing The Opportunity, and then Acting On It. There was no technical wizardry involved, and no resources that were not also available to thousands, and soon tens of thousands, of others.

The Jobs and Wozniak story makes this even clearer. Wozniak showed his designs to HP and offered it to them. HP's engineers were utterly uninterested in it. (And remember that the Apple 1 was built with off-the-shelf parts anyone could buy.)

There were few programmers at that time, so few programmers could have potentially reach the market. Compare it with now when you just need to download an open source programming language and compile it on many platforms.