This was suggested in August of last year, but I haven't seen any further discussion or movement. Currently, the only way to show elapsed time on something like a run, is to mark it as a "race". The timing data is already there in the analysis portion of the activity, so it's just a matter of giving the user the option to use that vice "moving time".
Link to the original idea:
https://communityhub.strava.com/t5/strava-features-chat/give-user-option-to-show-elapsed-time-not-ra...