Its odd to me that you think land-based transmitters won't tell you the altitude you are at. Why do you hold this quaint belief? It has nothing to do with the transmitters being in space. It has to do with how many there are.
Due to the error involved, unless they are plastered all over the place.
A transmitter directly to your side will allow a small error in the distance to result in a very large error in the height.
Lets say it is 5 km away, but it thinks you are 5 km and 5 m away.
That would be consistent with you being 224 m above a point 5 km away.
In order to provide accurate height data they would need to be very high (like in space) or there would need to be so many it isn't funny.
First off, your presumption seems to think each transmitter is at the same altitude, meaning not only a flat earth but one with no hills or valleys - oh my. Even given this absurdity, you are still wrong. They would still be able to determine the altitude.
No, in the case they are all at the same altitude and Earth was flat, how would you be able to tell if you were above them or below them?
The distance to each if you were 100 m above would be the same as the distance if you were 100 m below.
To show it even works with 4, lets assume there are 4 transmitters arranged in a square, with each side length 1 km.
You are 1 km from each transmitter. Are you above or below them?
The same situation occurs any time the transmitters you are detecting are coplanar.
And in fact, also happens in the general case when you have 1 fewer transmitters.
From a single transmitter, you have a hypershpere.
From 2, the intersection produces a sphere.
From 3, the intersection produces a circle.
From 4, the intersection produces 2 points.
From 5, the intersection produces 1 point and is overconstrained.
So until you get up to 5 transmitters you will have ambiguity in the position.
I'd like to know how he thinks it works on a round earth though, given his claim that there would be two solutions which would hold for a round earth or flat one.
If that happens with the RE, the point off in space is discarded.
The problem is with land based transmitters, unless they are all at ground level, which point should they discard?
If they are mounted 100 m high, should they discard the point at ground level or the point at 200 m?
There is no way to tell as you would want GPS to work for both.
With satellite based ones, should they discard the point at ground level, or should they discard the point potentially up to 40 000 km above the surface where the transmitters aren't even transmitting towards?
But ultimately, all of this is mute.
It doesn't matter if satellites are needed or not. What matters is the fact that they are used, and to calculate your position based upon the signals currently being transmitted you need to know the location of the satellites, which change.
There are open source GPS recievers and if you want you can just download the almanac/ephemeris from the satellites.
So you either need to have these satellites, or you need so many ground based recievers the Earth would be blanketted with them to fake the signals from the satellites for every location served by GPS.