I guess it's also possible via Ealing Broadway - Acton Town - Earl's
Court.

Yes it is possible, with or without a change to the Piccadilly Line at Acton
Town.

journeyplanner.tfl.gov.uk state that requires approx 10 min' more and
a bus interchange... not sure if can be travelled with a zone 3 only
travelcard...

You would need a 2+3 Travelcard in that instance as you would be passing
through zone 2 from Turnham Green to Earl's Court to East Putney.