Wireless Routers

You tweaked my curiosity.

This is from a Linksys Wireless Access Point manual

"Antenna Selection. This selection is for choosing which antenna transmits data, left or right. By default, the Diversity antenna selection, used to increase reception, is chosen."

So, from my experience with RF, here is your answer (at least in relation to this particular device).

Even though the antennas on a wireless access point are within inches of each other, the signal strength to a given receiving device can be very different from 1 antenna or the other. The Linksys device can be set to use either antenna exclusively, or to allow the device to select the antenna that has the best reception.

Now, remote connecting an antenna presents a set of unique problems.
First, for a remote antenna to even function, the coaxial cable connecting the antenna to the device needs to be of the correct impedance, and the length needs to be an EXACT multiple of the carrier frequency. At 2.4GHz, a cable length error of just a few millimeters can cause all of the RF signal to be absorbed inside the cable with no signal reaching the antenna.
Next, the antenna needs a suitable "ground plane" as part of the transmission path. A small metal plate will be needed just below the antenna.

In the days of CB radio, CBs didn't work well on cars like Corvettes because the fiberglass body was not a good ground plane.
The roof of a 75 Caddy or Buick was a much better ground plane than the roof of a Pinto.

A remote antenna will wok "best" if you only have 1 device that you use with the access point.
If you have multiple devices, the constant switching between antennas will impact system performance.

Bottom line:
No, you don't "need" both antennas on the router. If you run only one antenna, you need to set the router to only use the connected antenna.

While possible, it may not be practical to setup a remote antenna.

Chas