Sidelobes aren't really 'soft edges' to a main beam, they're unintended peaks at angles outside the main beam which sap gain. You can see peaks here around 29, 55 and 68 degrees:

https://www.gpsworld.com/wp-content/uploads/2020/10/Improved...

As such, successive GPS satellite antenna designs have been refined to reduce sidelobes, just as deep-space operators have realised they could be useful.

TLDR: sidelobes aren't like gentle feathering at the edges of a beam, they're more like petals of a daisy.