In a lot of places trains connect even small towns to larger cities. Not just a couple trains each week or each day, but coming often enough that you don’t really need to check a schedule.

A big part of the anti-car movement is being pro-infrastructure.

Being anti-car is more about being pro-infrastructure-that-does-not-rely-solely-on-everyone-needing-cars. It’s not even completely opposed to using cars/trucks for situations where nothing else will do.

My solution for this sort of arrangement would be to take the dogs and equipment for a walk down the street where a bus or light rail network would arrive every 2.5 minutes to take you to the train station. There, you play with or groom your dogs in a personal cabin on a bullet train that would take you to a similarily networked train station in the destination city.

Much of the required equipment would likely not be needed, since you could entertain your dogs on the trip rather than locking them away while you drive. If you still need a good deal of equipment, you could rent a car to get the station or hire a taxi cab.

Really unless you are making such trips daily, it wouldn’t make sense to need to own a car to transport a lot of stuff to the station in a nation with proper infrastructure. Just like you don’t buy a moving van because you plan to move at some point.

Out of curiosity, why do you take 10 dog crates for 4 dogs?

The trouble with ‘Slippery Slope’ and ‘No True Scotsman’ is that they themselves are not fallacies. Invoking them without proper justification is the fallacy. The same sort of thing happens all the time with ‘Appeal to Authority’, you can probably trust a scientific consensus about a subject in which they are all experts, but you probably shouldn’t trust an individual expert on a topic for which they are not recognized as an expert.

For an example of Slippery Slope: Fascists will absolutely try to demonize the most available target, and then because they always need an out-group, they continue cutting at what they consider the ‘degenerates’ of society until they are all that remain. (And then they find some new definition of degenerate)

“No True Scotsman” is valid in that there is at some point by definition after which you are no longer talking about something. “No true vegetarian eats meat” is valid, as this is definitional. “No true member of Vegetarians United eats meat” lacks proper justification, and refers to an organization, not a proper definition. This gets really messy when people conflate what group people are in with what they ‘are’ or what makes them a good example of a group. Especially when religion is involved.