The display/control board is a separate board that doesn't appear to have any smarts on it. The main circuit board has the micro controller on it. Looking at pictures, there are enough lines in the connectors between the main and display boards that the switches and LEDs are probably passed through as discrete signals. This makes the take over of the control panel the more attractive option, rather than trying to replace the main board and dealing with the high voltage/power interfaces. I'll assume that the functionality you would like can be implemented by manipulating the front panel controls in some fashion and at the correct timing. Then the question is "how far do you want to take this?"
If you want to maintain the front panel controls and functionality as it, then you can build an interposer connects in between the display board and main board. Signals coming from the display board are intercepted by a micro controller on the interposer board. They can then either be passed through or modified according to the new smarts being implemented on the interposer. The interposer board is what feeds actual signals back to the main board.
If you want to replace the entire display board, the your replacement just has to implement the same electrical interface as the old interface board. What you actually for the new interface board doesn't matter after that point. Want to put a touch screen on it? Great. Want to remove it completely and just use WiFi? That works too.
Figuring out the control lines between the display board and mainboard shouldn't be too hard. It looks simple enough that you can probably get away with just a multimeter. Follow the traces from the connectors to the various switches, LEDs, and potentiometers. There aren't enough switches for them to be have to be multiplexed, so that will simplify things. The seven segment displays will be multiplexed, and it's possible they chained the LEDs on to that, effectively making the LEDs on the board look like another one or two seven segment displays. http://www.ebay.ca/itm/Toyotomi-Toyosto ... Sw~G1ZyUbY
Depending on your programming chops and familiarity this would probably be good place for a Raspberry PI Zero W. Gives you wifi, blue tooth, and a full environment to run in. Pick your language, though Python is an obvious choice here. You could also use something a bit more dedicated, any of the larger Arduinos with a wifi shield would work. But, development would be a bit more invovled. The trade off is that you get much faster startup and don't have to deal with side effects of running a full linux distribution.