I have built a simple ground plane antenna for 70cm with a connector, 4 radials, and one vertical element. The RF ground is connected to the radials, and the RF is connected to the vertical. ... IF I went outside, connected one end to a deep ground rod, and the other end to the vertical element, would the radiation be the same?
Both conductors of a transmission line convey equal amounts of r-f energy, just 180° out of phase with each other. One conductor of the transmission line may have a conducting path to Earth ground at/near a transmitter, but that does not negate its ability to pass r-f energy between that transmitter and its load (antenna).
» An "r-f ground" does not exist at the distant end of a conductor having its near end connected to a buried, vertical ground rod. «
Probably the antenna system described in the OP already has a physical path connecting one conductor of its 2-conductor transmission line to some form of a nearby "ground" reference, such as the buried ground rod of the a-c service lines entering the building, and powering the transmitter.
Other things equal, there would be relatively little electrical difference in the performance of this antenna system if the transmission line conductor now connected to the common point of the radials of the ground plane antenna was disconnected at the antenna feedpoint, and reconnected there to a different conductor providing essentially the same function.