Why use a capacitor instead of just a wire?
The capacitor is there to defeat the modem control line serial auto-reset circuit in a unique situation where it is undesirable.

Arduino boards are normally programmed via a bootloader which runs immediately after reset, and then passes through to the application program or "sketch" if it does not hear programming commands on the serial UART within a brief timeout period.

Entry into programming mode has for many years now been accomplished by a small coupling capacitor from a modem control line, which is lowered by many host operating systems when a serial port is "opened", and also typically explicitly lowered by the programming software.  That pulses reset briefly low, resetting the chip and giving the bootloader a chance to run.  (It also typically means that people's "sketches" start cleanly over when a serial terminal program is launched to monitor them)

The asker briefly mentions that this large add-on capacitor is added when the Arduino is to be used "as a programmer".  In this case the software running on the Arduino should not be the bootloader, but rather the "Arduino as ISP" sketch, or application program, which causes and Arduino to accept serial commands and operate its SPI pins in a way to program a slave device.  Since the Arduino itself is not being programmed, it must not enter bootloader mode when the serial connection is begun by the host software.  Placing the larger capacitor defeats the small on-board coupling capacitor, and prevents the undesired automatic reset and entry into bootloader mode.

If a wire were used it would have to be placed to the positive supply instead of ground.  That is an option, though it could make power-on reset unreliable if installed before the board is powered.  A wire also has a greater risk if it is put in the wrong hole (for example, the next over is Vin, which could be 12v or so from an external supply) it would couple enough energy to cause damage, while a capacitor might not.  But a jumper wire to the positive supply can be successfully used with care.

It is also sometimes traditional to make the ISP sketch accept programming commands at a different baud rate than the native bootloader, to lessen the chance of accidentally reprogramming the "programmer Arduino" rather than the target, if an undesired reset and bootloader entry does somehow happen.