You have found the head of the screw
. Yes, a 2.8" has the exact size for the hole, but seems like it is needed to cut a little of the case to insert it. A touch screen seems to be a valid option and probably the best option because you can construct a module that is full independent of the number of buttons that the transmitters have. The module could work alone, and to be incorporated in whatever transmitter.
The problem is with the number of I/O. You need:
8 digital inputs for the digital trims or if the radio has analog trims, then 4 analog inputs.
4 analog inputs for the sticks.
1 digital input for the power switch.
6 digital inputs for the buttons at the screen.
1 analog input for the potentiometer.
For the switches. There are two switches with two positions, so 2 digital inputs. There are three switches with three positions, so 6 digital inputs. A total of 8 digital inputs.
So finally you need 8+1+6+8=23 digital inputs, and 5 analog inputs.
Also you need the I/O to control the RF module and the sound. Also you need the I/O to control the screen and if you want, the touch screen and the SD.
The problem with the bigger screens is that usually they are controlled with 16 data lines, the small screens can be controlled by SPI that need less lines. A minimun of 23+16=39 digital lines, plus control of the screen ¿?, SD ¿3?, touch ¿3?, RF ¿3? and sound ¿1?.
Really is needed a big number of I/O lines if you want a generic main board. Other option is to use a buffer for read the inputs, or to use and IC for the A/D but then you need to build your own main board.