From what is advertised the kit should work.
If the "Zero-Delay" encoders (can not judge if they are genuine or clones) put you off (e.g. being not so zero-delay) you can also run your setup with two MCP23017 (GPIO extender) and via GPIO.
If I would build your setup I would provide the left/right flipper a dedicated pin.
With the kit, as well as with MCP23017 you have 16 switches. Usually 4 are needed for the stick axes, and 9 for the most common push button functions: A, B, X, Y, TL, TR, Select, Start, Hotkey. So you have in any case three spare switches.
The third could be "Tilt" triggered by a gyroscopic sensor. (sometimes Space-key on keyboard based pinball games)
Mapping of the buttons can be done on a per game basis for many emulators -out-of-the-box- (retroarch-based and amiberry). And for any other emulator this can be done via runcommand-onstart.sh / runcommand-end.sh  in combination for example with xboxdrv.
Sorry, a lot to digest for a new joiner. But no worries, I assume you will master it by time.
TL;DR: Two separate pins/inputs for the flippers.