The two technical issues I'm aware of are the sheer number of satellites required for LEO internet, and the fact that you can't point your dish at a single place. There would need to be some sort of actuator or omnidirectional receiver for tracking to satellites at every client site. This makes the installation a bit trickier, but the ping times should be entirely reasonable provided someone gets the funding to put hundreds of satellites into an internet constellation.
For LEO constellations, each satellite can only see a small portion of customers at any time and will quickly move out of coverage of a single point on the earth, requiring many groundstations.
Alternatively, the satellites can crosslink and eventually hit a groundstation, but these handoffs and trip lengths quickly get back to the ping latencies of making a single trip to GEO.
Otherwise, with GEO/GSO satellites there is no way to get below ~280ms round-trip times.
The problem with existing high-speed satellite internet is that the satellites are all in geosynchronous orbit, which is about 22,000 miles up. For a roundtrip, your data has to go up to the satellite, down to the ground station, then back up to the satellite and back down to you, for a total of almost 100,000 miles, or more than half a second at the speed of light.