Driving is a privilege. Not a right.
The notion that "X is a privilege, not a right" is something that you often hear from laypeople, but lawyers and courts don't use the words "privilege" or "right" to make this sort of distinction, and I'm not aware of any modern Supreme Court doctrine that depends on classifying something as a "privilege" versus a "right."
Indeed, the "right" of interstate travel that everyone in this debate seems to agree exists (i.e., the right to travel by some means, not necessarily by your preferred means) is one that the majority of the Supreme Court in at least one case has suggested is based upon the "Privileges and Immunities" clause of Article IV of the Constitution.
Zobel v. Williams, 457 U.S. 55, 66, 73 (1982). In other words, the majority of the Supreme Court in that case was willing to accept that a portion of the Constitution that guarantees "privileges" created a "right to travel."
Perhaps those on this board who are so convinced that they can answer a debate by saying "privilege, not a right" should explain to the poor benighted Supreme Court that it has been reading the Constitution incorrectly.
As for what the cases that you cite actually held,
Miller v. Reed said that a man that refused to give his social security number to the DMV to renew his driver's license because he claimed that sharing his social security number violated his religious beliefs had no right that the state was violating.
Dixon v. Love said that the state could temporarily revoke a driver's license for three speeding violations in a 12-month period prior to holding an evidentiary hearing (the driver argued that the revocation should not have happened until after the hearing). In
Berberian v. Petit, a driver whose license was revoked for a second DUI challenged the revocation because it was made by the state registry of motor vehicles, not by a court, and the court said that there was no violation of the driver's rights.
Really, none of these cases was about the right to travel.
Miller was really a religion case, and both
Dixon and
Berberian were due process cases, where the question was not about whether the right to drive could be taken away for repeated speeding or DUI (the drivers in both cases appear to have conceded that it could be) but about the level of procedural protections the state needs to provide before it takes away that right.
As these cases show, the cases where people argue for a "right to travel" by car tend to be brought by unsympathetic drivers, that are arguing there can't be any rules at all, have demonstrated repeated unsafe driving behaviors, or are claiming implausible religious beliefs. For understandable political reasons (most voters depend on the ability to drive to work and/or carry out basic tasks in life), governments in the U.S. very rarely try to impose unreasonable restrictions on the right to drive. If a state really tried to enact a crazy restriction on driving (e.g., nobody can drive in the state on Tuesdays), I suspect the courts would find a way to conclude that there is a right to drive a car.