Google Maps is very clearly "designed in California", with no awareness of the rest of the world.
I had the same impression in Asia, where subway exists are numbered and it makes a lot of sense to get out at the right exit but Google Maps does not really take the exits into account although they are on the map.
It also does not display the non-Latin names of stops, which would've been useful sometimes, especially when asking people for directions. People in Asia aren't often terribly familiar with the names of the stations in English.
Did people wake up one day and find that their "Main Street" was now just "part of Route 66, it doesn't have a name" ?
The letter-number designations like "A33" in the UK are like your route numbers. Some were purpose built and nobody cares what the underlying road is "named" (all the major Motorways are like this, there is no "name" for the M25, it's just the M25) but most are pre-existing roads, designated as part of some route and to locals the original name is what they'd call their bit.
e.g. Nobody where I live would say "the A335" they'd say "Thomas Lewis Way" because that's what this part of the A335 is named even though it was constructed specifically in order to bypass local streets for the A335 route. A person from out of town might say they used the A335 but probably people would look puzzled, then go "Oh, Thomas Lewis Way, yeah".
You're right for practical purposes, but they do often technically have names. The M25 is the "London Orbital Motorway": https://en.wikipedia.org/wiki/M25_motorway
In the Midwest, at least, it is very common for larger roads to have both local names and route numbers (one of county, state, or federal); no, the road name is not removed that I've ever seen. An example, heading due west from Chicago is state route 38, which is Roosevelt Road through the city and suburbs, then has other names as it passes through smaller towns - often "Lincoln Highway," as 38 generally follows that historic route. Frequently state routes will have the name "State Street" within smaller towns.
The Interstate system is a departure from this, and rarely do they have names - although in Chicago they do have de facto names, which often confuses outsiders when they hear the traffic report. "What the heck is the Ike?!"
Overall, this is complicated and it's not surprising to me that it's hard to get a computer to do the right thing.
Not really. In New Jersey for example, almost all major roads have both a name and a number, and both are displayed on Google Maps.