Cyclists can ride extremely close to other cyclists and remain totally safe.

No, they can't. The risk of a careless cyclist swerving across the path of another cyclist is little different to the risk of a careless cyclist swerving across the path of a motor vehicle, except that in the first case the innocent party is much more likely to be seriously hurt. The point of the official guidance to allow as much space when passing a cycle as when passing a car is because the cycle is more likely to be unstable, not because the overtaking car is.

Cars are only a problem in high traffic routes with a large need for passenger throughput. Buses on those routes tend to be well utilized.

Several decades of experience living in my current area suggest otherwise. For example, one significant problem is asymmetry: if you go one way on a journey by one mode of transport, you may well need to make a return trip later using the same mode of transport, but just because it's busy and there's a frequent and well-utilised bus service in one case, that is no guarantee of the same (or indeed any bus service at all) for the return trip. This can simultaneously result in both hugely wasteful journeys by almost-empty buses in some cases and impossible journeys due to inadequate bus service provision in others.

If you are living in a place that doesn't have that quantity of people passing through (more rural areas) then cars are totally fine.

But people living in rural areas often travel into or via nearby towns and cities. Indeed, it's entirely possible for the majority of congestion on the roads of a town or city to come from this source. Unless you provide adequate public transport alternatives for not just those travelling relatively short distances within that town or city but also those travelling into and/or out of the central area, you don't really offer a practical alternative to car journeys.

Likewise, people living in big cities need to travel outside the city sometimes. Unless there's reasonable public transport for their entire journey, including reaching any long-distance transport in their home city and getting around wherever they are visiting, again you're back to needing an alternative that can do the whole job, and if that means a car then you're driving it within your home city as well.

The issue is that, in busy cities, clear roads are an impossibility, even in cities that were built with automobile traffic as the clear first priority for transportation.

Again, that is not necessarily true. When you reach critical mass, it becomes true, but there is often a wide band with some degree of congestion but not approaching gridlock. Within that band, it is entirely possible for journeys to be significantly disrupted and therefore lengthened by the effects I mentioned. We see it almost every day in the city closest to me, for as much as an hour or two either side of peak periods, or during lesser peaks like weekend shopping or before and after some local event with a lot of visitors.

The car free movement is fundamentally about cities (particularly city centers), where cars are inefficient, are expensive, are dangerous, use up in inordinate amount of space, and are a major contributor to pollution (which is a particularly large problem in such areas). In those sort of situations, there is simply a better way.

Yes, mass transport is effective in scenarios where there is sufficient demand to make it worthwhile, and for local journeys within dense populations at peak times that may well be the case. I don't think anyone seriously disputes this.

The problem I have with a lot of the anti-car advocacy is that it treats transport networks as some sort of boolean system, either "congested" or "not congested", but in reality the degree of congestion tends to follow a much smoother curve with peaks at various times throughout the day. If you draw a straight horizontal line, across a chart of congestion against time of day, at the level where critical mass is reached and gridlock is inevitable, in many places there will be a lot of the time when the curve is below that line. At those times, alternative modes of transport may be less attractive in terms of potentially relevant criteria like efficiency, reliability/availability and environmental friendliness.