As with every project in the Google age, I started by, well... Googling it.  Apart from the overall design, there exists no identifying marks on the chime, except for some indication of the required voltage etc underneath the volume knob.  So after some creative googling (doorchime 3-note, chime old, chime 1950s , etc.) I was able to find the following reference:

https://www.electrachime.net/resonator-door-chimes/rittenhouse-beverly-1949/

Beverly Model – The highly popular model is ideally suited for those people who prefer a chime which takes up less space than the long-tube model. It has a beautifully rich resonant tone, a melody of either 3 or 4 notes. A simple link attachment makes it possible to install the chime to play either a 4-note melody or a shorter 3-note sequence if the owner prefers. The Rittenhouse volume control permits adjusting the tone to any desired level. A shut-off switch also makes it possible to completely turn off the unit if for any reason you do not wish to be disturbed. The rich, ivory plastic housing and the graceful brushed brass tubes combine to provide a beautiful design that is in good taste in any home.

With the model number in hand, I was able to narrow my Googling, and ran across the delightfully helpful site Knock Doorbells.  Here is where I was able to learn pretty much everything I needed to know not only about how the chime worked, but also about the inherent dangers of this type of chime.  Originally I was just looking to replace parts until it was in working order, but with the knowledge gained from Knock Doorbells, I decided it was worth trying to upgrade it in order to minimize my fire risk and give it some new life.