Offering a GPS-like service would require totally different hardware. Each GPS satellite has 4 atomic clocks. The StarLink satellites will basically be optical routers and their timekeeping needs can probably be handled by a cheap clock and NTP from the ground.
Could have multiple clock sources on earth that produce the clock signals for the satellites. So a satellite could get an accurate time if it locked onto a few ground based sources.
Just with that volume and density of satellites, kinda begs some lateral thinking.