Drivers are shown off to the crowd at the track before every race in the driver’s parade where they’re also interviewed. They’re also interviewed before and after every practice session, every qualifying and every race and that’s not even including press conferences, channel specific coverage and the open access to their radio transmissions. There’s no way they’re anonymous. Even someone like Brendon Hartley, who’s only doing that one race, was far from anonymous.
The other problem is that the time it took cut into the driver’s pre-race prep - they were standing around in the heat for 10 minutes and then had to hurry to their cars. It also removed drivers from being available to broadcasters when they usually do interviews on the grid, which is often when you get drivers at their most candid and get an insight into how they and their cars are immediately before the race.