I like the DIY attitude, and I hope you’re able to solve this one yourself, because I like the feeling of accomplishment that comes with that, but I also understand when people don’t want to spend the time and would rather have someone else do the work.
I’m guessing this means you didn’t rewire the current chime to its original condition and test with the “dumb” doorbell button, and I can’t say that I blame you for that. It’s something I think I might do to help myself think through the problem.
When you get the working multimeter, you can use that to test voltages at your transformer, at the chime location, and at the doorbell (depending on what you do with the wires in the chime box) and you can also use it to confirm wire runs and determine which pair originate and terminate at which location, which is what I found helpful and one of the things I accomplished today (Steps 6-8 in my post from earlier today). (You might already know all of this stuff. I don’t know what your level of experience is. I’m just learning as I go and trying to share what I’ve learned with the hope that others might find it helpful.) Being able to do my own tests and see the results of those tests has really helped me to have a better understanding of the wiring in this house and how it’s supposed to work.
Now I guess I just need to keep 
and see if the chime continues to ring after the mucking about I did today. I have good feelings about it.