An update from my USB-C replacement: it turns out that the USB-C replacement did not fix my MacBook. I found that if I powered it down then plugged it into the charger, it will charge.
After replacing the USB-C port and plugged the computer into the charger, I did not hear the usual tone that lets me know that it is charging. I decided to leave it plugged in to see what happens. I also did SMC clear (Shift + Control + Option + Power) for good measure, but that did not appear to make a difference. After several minutes of charging, it made a tone, flashed a battery symbol on the screen, then booted.
I didn’t notice this when removing the USB-C port, but it seems to be a little loose under the screws. I have another MacBook of the same vintage and found that its USB-C port also has a little movement when a cable is plugged in. Not much movement, just a little.