A number of early players of the Telecaster in the USA played them with the bridge/pickup cover on- the late great Jimmy Bryant , and a number of black r'n'b players like Guitar Slim, Albert Collins and others..I think Neil LeVang and Buddy Merrill played their Strats with the vibrato bridge cover on too in the '50's...
Also, most early Fender bass players left the armrest/pickup cover and bridge cover on too.
All of these people probably didn't question why the fittings were there- they probably figured that Fender put them on there for a good reason! Mr Fender's ideas were usually functional and musician friendly- I bet your hand wouldn't get shredded by the Strat bridge screws if you could stand playing with the cover on, and the bass arm rest would also protect the pickup from accidental damage..
It was only in the mid-late '60's that players started removing all the hardware- because it was thought to look 'cool' perhaps--there was a similar contemporary Clapton/Beck inspired trend to take the pickup covers off Gibson humbuckers too...it was supposed to give more power and sustain or something..