The compelling reasons 'patio11 offered were that it will cost a lot of money and years of opportunity cost. If you don't find those compelling, then that's your prerogative.
The whole NASA or Intel thing... I think you hit the nail on the head with "building valuable connections". I think you need to do this regardless of where you want to work. If you decide you want to work at one of these places or some place like it, make sure that you understand what is and is not required for the job. A certain degree may be required (explicitly), there may be age expectations (most likely illegal, but it happens), and/or there may be soft skills that you may not have yet and may never acquire (a very real requirement to get hired that is often overlooked by many people looking for specific jobs).
FWIW, 'patio11 has some high quality experience with the engineering hiring market and has written some seminal texts about it. Rather than blithely reject his advice (as you seem to do here), I humbly suggest that you engage with him in a more constructive way, perhaps by engaging with questions (e.g., about his thought process, about specific details on career direction that you failed to mention in your original post, etc.).
Also note, as a general rule, people with highly-informed and high-quality advice typically aren't looking for an argument, a debate, or (worse) a moving target of discussion -- you seem to have taken your reply one or more of these directions. The way you engage with people matters, and I wonder if this is a limiting factor of your career development.
Regardless, best of luck in your search. I hope you keep us updated with your status.