Because if you need to explain GPS at the level of the impact of general relativity, my understanding is that by itself is already a topic normally introduced in the final year of a physics degree.
If you're OK with simplifying to "time passes at a different rate for the satellites, here's the equation, I will not explain why it works just roll with it", why insist on ceasing to use it if the foundation isn't understandable?
What you've suggested doesn't work at high resolution at sea or in the wilderness due to lack of radio towers at suitable frequencies, atmospheric variations and ducting influencing the signal of the existing systems.
Doesn't work in valleys or cities due to multipath reflections.
Only works in low rise areas or while flying at high altitude.
That's why the satellites got built in the first place, and why China and the EU added their own satellites to the US and Russian global navigation satellite system.
It is more funny than I initially thought: We keep making things more complicated and secret in order to make something else less complicated. Even the luddites accomplished only to make things more complicated.
I found a similar treasure some years ago: How we take convenience for self-evident. We love to pretend everything should be as easy as possible. You should really never have to think about anything, never have to learn anything, never do anything, nothing involving muscles, endurance or fine motor skills. Life is best without developing and without accomplishing.
Any human technology that the general population can't understand is an egregious failure in education that will be indistinguishable from large-scale cognitive impairment. -- h-d'a
There has never been a time when most technology in use was understood by the general population.
"The Last Man Who Knew Everything"[0] was 1773–1829 and the trend towards compulsory education was only beginning, 2 countries at the start of his life and 7 at the end[1].