If you're crossing a land border into the US you can't be "refused travel" - if you don't have the ESTA you won't be allowed to board a plane to the US but if you present yourself to immigration at a land border you don't have the issue of trying to board a plane. Whether you would be allowed to enter at all without ESTA is another matter.

It's better to have your ESTA registration seen to beforehand. The last thing you want on your round-the-world cycling trip is to get refused entry to the US when your flight leaves from Canada - that would leave you trying to find a flight from Mexico into Canada without a stopover in the US, and that word Rejected in your passport can (from what I gather) cause you problems for up to 10 years.