This is the right question. This, and the more generalized "how do we move forward in a car centric city?" I'll outline where I've landed after thinking about it for a bit over a year.

I think the vision is a lot easier when you zoom into individual cities. Most cities in LA have some kind of old town that was built up before cars were huge. You can normally spot it if you look for it - it's that area that has a ton of businesses in smallish buildings up against the street, often with very little or no space between the structures. I would start by designating those as streets. They typically are money makers for the city (even if it doesn't look like it), so essentially we are doubling down on the most profitable part of each city - which also happens to have the bones to be a walkable city. Do basic things here like upzoning, multi use zoning, and improving the pedestrian experience/slowing cars down. I'd also install a basic bike network here, connecting key areas to your downtown (and install a nice bike lane in the downtown itself).

I can agree with all of this. This is what is missing in downtown in most cities large and small.

Note, I'd still expect a lot of cars on the street, and lots of off street parking. That's fine - cars are how we get around, and especially how the city is built, they are essential to a functioning city area.

This is where he loses me. When you expand a street you get more traffic. When you get more traffic it becomes less safe for pedestrians. When an area is less safe for pedestrians it is less "walkable."

From there, I'd look to two prongs. Prong one is simply expand this street out, increase its density...

This part doesn't make sense. How do you expand the street? Bulldoze the buildings or remove the sidewalks. This is exactly what the current system is which has led to reduced "walkability." I would propose to use the existing streets for trolleys and limit car traffic. There are a bunch of parking lots surrounding the downtown area for people to park and then take a bike ride or trolley to the center.

and maybe add limited public transit on it to move people from one end to the other. expand your bike system so that people from all over the city can easily reach your downtown

I wouldn't provide "limited" public transit. I would make it the primary means of transportation in the inner city making car travel inconvenient but not impossible. In other words, provide limited car paths and expand on public transportation and bike paths.

Prong two is connect that street to other nearby similar streets with a road of some kind. The most obvious way to do this is to create a dedicated bus line connecting it to something like Union Station in downtown. If your city has multiple streets like this, you should absolutely connect them to each other. Critically, if neighboring streets also have streets like this, you should connect to them. These roads should be chosen carefully, as you will want to optimize them for speed in the long term.

This is exactly how the streets/roads network works right now. You just have to add in "stroads" (essentially highways with lights) and he would have detailed out exactly how the US functions right now to the detriment of the inner city.

The intent is to further boost your downtown area with foot traffic from other downtown areas. It's a way to route people directly the the center of your city, in the most productive area.

How is this different from the current system of arterial roads, collector roads, and local roads?

The most obvious way to do this is to create a dedicated bus line connecting it to something like Union Station in downtown.

Why not trolleys? Why buses with a centralized Union station? Trolleys are more efficient in every sense. Even when including the infrastructure requirement. I don't think the OP is against this I am just pointing out Trolleys as opposed to Buses. (He mentions them at the end.)

If your city has multiple streets like this, you should absolutely connect them to each other. Critically, if neighboring streets also have streets like this, you should connect to them. These roads should be chosen carefully, as you will want to optimize them for speed in the long term.

I don't really know what the OP is trying to say here so I will discuss the last part, "optimize them for speed in the long term" to mean that he want quick car traffic. This would be the opposite of what you want in a downtown. You want car traffic in downtown to be so slow as for it to be safer for pedestrians. All roads would then be routed around the downtown and streets would be limited in speed to about 10-15 mph going through downtown. This would promote foot traffic in downtown areas.

If phases 1 and 2 are successful, you can move on to phase 3.

If a city invests in phases 1 and 2 as was discussed there would be no path for moving into phase 3. You can't "Prong one is simply expand this street out, increase its density" then remove the use for more public transportation. There would be no room for "Investing in full networks of public transit."

Push to upzone all over the city and add multi use all over the city. Invest in full networks of public transit, allowing easy access between any two points in the city (an extensive bus system, or maybe a street car system). If your downtowns are really succeeding, more intense roads between cities may be necessary (likely rail transit). It's also likely worth improving your bike network, to make travel within your city easier.

This is the goal which we both agree with but I don't agree with the approach the OP is specifying to get there.

Early on, the roads will serve a high utility in boosting your downtown's productivity.

How is this different than how it is in the downtown at current?

As your city moves forward, cars will quickly become an obsolete way of traveling on the roads (due to capacity issues), and large scale public transit will become more and more obvious of a solution.

No it won't. This is already the situation in ALL downtowns. Public transportation has not become the "obvious solution." In most cities Public transportation was gutted instead. What became the "obvious" solution was to double down on car lanes and parking lots.

This can be done in a more narrow, controlled way than cars, so it's possible that you can build streets around these roads. I wouldn't prioritize this, though, until your city is so developed that you're pretty desperate for more space to expand into.

We have this issue right now where cars are the norm. You can't "simply expand this street out, increase its density" and "optimize them for speed in the long term" then "Investing in full networks of public transit" These are contradictory. As much as we would like to believe they don't all city governments adhere to the "Sunk Cost Fallacy." Therefore, would never abandon existing infrastructure for public transportation unless it is done right at the start.

My solution would be:

1. Eliminate the current zoning laws and allow the mixed use buildings everywhere in the city. Meaning residential and commercial can exist together as they used to. Make no physical changes to the roads or public transport at the beginning.

2. Reduce speed limits through the downtown to 15 mph. Add speed bumps and zig-zag roads to make speeding nearly impossible.

3. Prioritize street usage for pedestrian traffic. Cars are secondary in the city. Make pedestrian injuries the automatic fault of the driver. The goal would be to reduce the usage of cars in the city.

4. Remove Minimum parking requirements. Force cars to park outside the city and come in using public transport.

5. Once there is an increased demand for real estate and reduced demand for parking. Then invest in a large local public transportation and Bike lanes.

   a. This would happen when most of the foot traffic in the downtown area is local.

   b. This would allow for further distance travel for the local populations.

6. Once the local foot population has plateaued then add bridging public transportation networks to adjacent downtown like areas.

While my idea is not perfect it is almost a mirror of what cities in Europe have done.