At first I couldn't understand why someone would use a separate A/D converter when most of the PICs have them built-in. The answer was in the code: it was written by a Maxim guy to demonstrate the Maxim A/D converter. Definitely the hard way. The code seems to support only one axis.

The PIC16C54C shown in the schematic is an OTP meaning "one time programmable". You have one chance to get it right or it goes to the landfill. They offer UV-erasable windowed parts for code development but you need a UV eraser to use them. The '16C54 is the oldest surviving member of the PIC family. Newer PICs are flash-based and can be reprogrammed many times.

If your heart is set on the '16C54:

Erasable = PIC16C54C/JW
OTP = PIC16C54C-04/P

My heart isn't set on anything except getting an inclinometer. Can you recommend a flash based PIC with on board ADC that can fit in this schematic or one like it?

Here is where engineers get their reputation for being "difficult to deal with", "not a team player", etc. Pointing out all of the alternatives and their consequences is often interpreted as making things hard.

Option 1: Use design as-is.
This is possibly the quickest path to a goo-sensor based inclinometer because the hardware and software already exist. This is assuming that the posted code is correct. One of the cons is that not all hobbyist device programmers support the PIC16C54. Another is the OTP aspect mentioned earlier.

Option 2: Replace PIC with a flash-based A/D PIC.
A more elegant but more difficult solution. The PIC16F818 includes the A/D converter, but you will need to write the code to run the A/D converter. Since it is flash based it can be reprogrammed without an eraser. Most of the '16C54 code will drop in. This option is only valid for someone who wants to master the PIC instruction set.

Option 3: Lose the goo-sensor.
They work great on the bench. Interpreting a signal from a sloshing vial of goo in a moving vehicle is another story. I'd hate to see all of your effort result in something that doesn't do the job.

"Goo sensor"
Sorry but that just made my day.