From the toy model it seems like if in addition to distance you could also get your angle to the landmark then you could get your position from a single landmark. Is that something which is practical? Knowing which angle a radio signal came from? It seems like if you had several receivers you could use the timings of when they receive a message to determine the approximate direction of the satellite in the sky. Given receivers have gotten much cheaper over time, is this a viable extra constraint to improve accuracy?
Determining the angle of a microwave signal requires a pretty big receiver dish (or an array of receivers, at least).
The wavelength of the GPS L1 band is 19cm, you'd need (roughly) something of at least 10x that size to get a good sense of where a signal is coming from, so on the order of 2m.
A single good antenna and sensitive amplifier gets you a fix on more satellites, which provides good accuracy with less space and hardware used.
Assisted GPS in phones can augment GPS data with information coming from cell towers nearby. From a quick search, some are omni-directional while others operate in a specific direction. Depending on reported signal strength, you could get some hints on the angle as well.
