If you recognize the property to be 'public' then you acknowledge the legal arrangement whereby the public makes rules for using public property. So driving around in a dangerous vehicle is regulated on public property. If you don't acknowledge public property, then all property would be private, and you'd only have rights on your own property. On others' property you'd have to follow whatever rules were established for uses.
As for freedom of assembly, driving an automobile has nothing to do with assembling. Next we'd have to pay for your cab to get you to the assembly. Walk if you want to assemble. I'd agree with your right to walk on public property.
The theory that you own that spot you occupy is silly. If everyone owned the spot they occupied on every road, then there wouldn't be enough room for everyone. It's just silly.
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