This is a pretty poor argument to be honest. For a start you're misrepresenting the claim.

Cyclists are very aware of how car-centric the infrastructure of their country is and the vast majority of them are arguing for resident focused planning, along with cycling infrastructure that allows people who aren't well-off to do away with the expense that is owning a car.

You can't purposely design everything around owning a car and then use "poor people need to own a car" as an excuse against any kind of mass transit planning.

You’ve got it backwards: Most people who ride bikes do not fit this description. Working class riders represent the large majority of riders globally. They ride for transportation, not pleasure, and on whatever bike they can, in whatever state they can get it. Because you’re right: a car is actually not an achievable purchase for those in precarious financial situations. This is the case for most forms of alternative transportation. So funding support for those forms is really important, even if for some reason you want to not do that to spite… moderately well off people.

This is what normal people understand right now when someone mentions bicycles: "Portlandia bike clip" [1]. It's a 10+ year old clip but all the better for it, sitcoms like Portlandia were in the vanguard of a lot of class-suppression and gentrifying stuff that we now see as "good for everybody!", not just for the middle-classes from which we're a part of (and which those sitcoms were targeting, self-parody used to be part of our class-related shtick).

[1] https://www.youtube.com/watch?v=V3nMnr8ZirI

You’re right, there’s a long way to go. My family think I’m nuts any time I ride my bike more than a mile or two, when I go get groceries on it, etc.

This view omits a lot of nuance. Few people are calling for an immediate, unconditional ban on cars, unaccompanied by other reforms. What's clear -- and well backed by research -- is that cars are the worst form of travel compared to the alternatives. The key, then, is to make those alternatives more realistic by accompanying a ban on cars with a serious increase in investment in better public transit and urban design.

This approach is far better for those who aren't "well-off". Cars are very expensive! If we can free people from that economic burden and reduce the environmental impact and improve the availability of alternate means of transit, then the call to eliminate cars is extremely well justified. An argument which, again, is backed up by a wealth of robust scientific research covering all aspects of the problem, including economic.