"The constitution is quite clear, people are allowed to block public thoroughfares to protest."
Which part of The Constitution would that be? First Amendment protection of "peaceable assembly"?
I would disagree.
One of the fundamental ideas of liberty is that your liberty ends at the point where it infringes on the liberty of others. Restricting someone's freedom of movement seems like an infringement.
When you block a road, you've essentially imprisoned people in their automobiles. They obviously can't proceed because you've blocked the road. On many roads, they would not be able to turn around either, so their only escape route is to abandon their vehicles. You do not have a "right" to put people in that situation as a means of protest.
Taking it a step further, if you believe that you have a "right" to block a public thoroughfare, can you therefore block a public street and sidewalk to prevent a person from leaving their home? Could you surround one or more people on a public sidewalk and prevent them from moving at all?
"If police deem it a problem, they can arrest them."
If the matter is left to the discretion of the state or local police on the scene, the only relevant parts of The Constitution would be the 9th & 10th Amendments. That would mean that there is no Constitutional Right to block a public thoroughfare & it would be within the purview of the states and localities to restrict the practice.