Does appear that U1 (CPU) is not original since the one in the photo appears to be black-topped. Could also be a source of erratic behavior.
Can you get a readable closeup of him?

That would be a 68xx series part with ON Semiconductor logo.
Common mistake made by counterfeiters - even though Motorola became ON Semi, these parts were never made with the ON Semiconductor logo.

These parts were put on Last Time Buy in 1999 with deliveries into 2000 (Product Change Notification 3932).
And Motorola Semiconductor changed the name to ON-Semiconductor in 1999 - about the same time that the last time buy was issued.
Final 68xx series parts were made with Motorola logo, it took a year or so before ON-Semi was putting the ON logo on parts and didn't put the ON logo on these discontinued parts.

The rough tooling holes is another thing to look at. Although some manufacturers had rough tooling holes, all of the 68xx series parts have had smooth tooling holes (see 6821 directly below it).

Yes - it is defintely a counterfeit.
Now was the part beneath the counterfeit marking a real 6808? Maybe yes, maybe no. We will never know.
Most probably this was a 'reclaimed' part. This means it was removed from an old board - original part marking was sanded off and then the counterfeiter remarked it as a new part. Due to how their reclaiming process works - these are often damaged internally.

Can it be causing the wonky behavior? Yes, it can easily be a source for wonky behavior.
Unfortuntely, these parts are long obsolete and the source for unused parts is practically non-existant. Your probably going to have to resort to ebay for these. A bit of scrutiny helps. Doing a search for MC6808 yields a list of 16 results excluding parts specifically listed as an "item found from eBay international sellers". Not sure why Chinese sellers aren't included as an international seller. Out of this page of 16 -- only 3 appear to be the real thing based on photos.
#1 is from Bocasemi at $25 each.
#2 is listed as Zebra MC6808P from Radwelloverstock at $5 each (I have no idea why "Zebra" is in the name) item 133466228657
#3 is listed as Zebra MC6808P from plccenter at $17 each (Again, I have no idea why "Zebra" other than copying Radwell's listing)

Be aware that some sellers show photo of one item but sell one that is different so take a careful look at it when it arrives if you order one of these. Right off the bat - it is obvious that plccenter stole the photo from Radwell as well as the description. If it were me - I would give Radwell a shot and see how it goes.

I don't know what Action Pinball or Marco Specialties have for parts. Before buying I would ask them about part as well as date code (biggie) and see if they can send a photo of the part.

Hey, try asking whoever you buy them from:
1 - manufacturer and manufacturer logo
2 - manufacturer part number
3 - date code (bottom line)

Then post here. We can often make sense of these to see if it is a possible good part or way out in left field.

This one had my curiosity up.

One thing that I ignored on these were the RE and Standby pins.
Pin 36 is "RE" or "RAM Enable" on the 6802. If low, it disables the internal RAM. If high, it enables the internal RAM.
I assume this does not exist on the 6808 but I'm not sure as my antique Motorola databook doesn't have the 6808 listed in it.

The internal 128 byte memory block is mapped to locations 0x0000 through 0x007F. If the internal memory is enabled, the CPU ignores external memory mapped to that location and only uses internal memory (if RE is high). Other than just adding extra loading to the CPU - any duplicated memory shouldn't matter for a 6802.

Never knew what the standby input (pin 35) did before. The 6802 has 32 bytes of RAM that can be battery backed up. Bally didn't make use of that and just tied the standby to VCC (enables this RAM but is not battery backed up). This backup capable memory was at locations 0x0000 through 0x001F.

Based on when these parts were produced - I would guess that slochar is right. Memory was very expensive back in early 1980s. The 6802 was the primary part but cost more with the included memory. The 6808 was cheaper since memory was not included (or included but not advertised). Many of the 6808s were probably made from 6802s when the 6808s were in short supply or when the internal 6802 RAM had issues.

Counterfeits.

1 -- Motorola never made these with the OnSemi logo.

2 -- The bottom line for Motorola parts include date code. Date code says "1438". 38th week of 2014. Pretty good considering these were discontinued in 1999.

Now what these were before they were blacktopped and remarked is anybody's guess. May have been 6802 or 6808s.. we'll never know. But the remarking of parts to purposely misrepresent them qualifies them as counterfeit.