power loss seems like a non-issue. there is no reason for a power loss unless this is not connected to a utility grid.
however, a simple battery can power the logic of a controller very easily.
I thought that bowling lanes were built in pairs ?
the schematic looks like one lane but the has 5 groups of 11 sensors. 1,2, 4,5,6
each labeled for a different part of the machine, so this is for one lane.
as a note, electronic does not have to mean no mechanical things. a simple gray scale encoder will tell you the position and you can increment that like a watch escapement. not uncommon for assembly lines to count each passing thing.
a simple NANO can be used as the Arduino replacing the counter. you get about 20 input/output pins.
the MEGA ups that to over 50, the ESP32 generally has 34 GPIO
there are ESP32's with an on-board battery charge/backup so they can run when there is no power (solar charge) or run between power loss.
You had mentioned re-setting. I assume when your count is to, say, 4 and there is a failure, you would have to cycle the entire machine to get back to 0 for all the other counters. Interesting logic to say the least.
the beauty of your machine is that you can just add a sensor on the unit, in front of the existing counter and let them both work independently. you get to test and address issues.
one thing I would think you could do is to position the sensor so that it sees the center mass of the pin. a rising edge of the sensor detecting pin would be followed by a falling edge after the pin completely passed.
I would try an IR sensor as well. just for fun. the beam break would work, but it seems that an IR would also be able to detect a pin.
once you get into the programming. you would see the sensor change state (detect a pin) then you can count up to some time (1 second) check if the pin has cleared, then save the count, get ready for the new pin
to address the voltage, you might be well served to have opto-isolators to create a barrier between the high voltages and the low voltages. simple to use, not expense and you can/should use simple indicator LEDs to watch what is going on.
the cost of custom circuit boards in so low these days it seems like you can make a custom board and have it stuffed with parts, probably for under $100, well, maybe quite a few for $100, it's like the first one is $100 and then next 4 are free.
Add to that the sensors and the Arduino board, stuff it all into an enclosure and you start moving into the electronics world.
