Well, it is the sovereign citizen crowd that typically makes the argument that the state is barred by the federal constitution from imposing the requirement of a driver's license. But it is and always has been a losing argument. The courts have made that quite clear, if you actually look at the cases in which that specific issue was raised rather than, as is typical for those making the sovereign citizen arguments, resorting to cases and laws that are not on point.
And as you mention California in your post, the Ninth Circuit Court of Appeals (which is the federal appellate court that covers California) explained why a driver’s license does not violate the right to travel:
We have previously held that burdens on a single mode of transportation do not implicate the right to interstate travel. See Monarch Travel Servs., Inc. v. Associated Cultural Clubs, Inc., 466 F.2d 552, 554 (9th Cir.1972) (“A rich man can choose *1206 to drive a limousine; a poor man may have to walk. The poor man's lack of choice in his mode of travel may be unfortunate, but it is not unconstitutional.”); City of Houston v. FAA, 679 F.2d 1184, 1198 (5th Cir.1982) (“At most, [the air carrier plaintiffs'] argument reduces to the feeble claim that passengers have a constitutional right to the most convenient form of travel. That notion, as any experienced traveler can attest, finds no support whatsoever in [the Supreme Court's right of interstate travel jurisprudence] or in the airlines' own schedules.”). The Supreme Court of Rhode Island in Berberian v. Petit, 118 R.I. 448, 374 A.2d 791 (1977), put it this way:
The plaintiff's argument that the right to operate a motor vehicle is fundamental because of its relation to the fundamental right of interstate travel is utterly frivolous. The plaintiff is not being prevented from traveling interstate by public transportation, by common carrier, or in a motor vehicle driven by someone with a license to drive it. What is at issue here is not his right to travel interstate, but his right to operate a motor vehicle on the public highways, and we have no hesitation in holding that this is not a fundamental right.374 A.2d at 794 (citations and footnotes omitted).
Miller does not have a fundamental “right to drive.”
Miller v. Reed, 176 F.3d 1202, 1205–06 (9th Cir. 1999).
Other circuits have reached the same conclusion. Indeed, it's funny that you cited one of them but completely mischaracterized it by not actually quoting the holding of the case which clearly states that state driver's license laws do not impermissably burden the right to travel:
But Dean has not articulated reasons to support his unexplained argument that state licensure and registration requirements violate the right to travel, see Fed. R.App. P. 28(a)(9). This is not surprising because such an argument is meritless. Miller v. Reed, 176 F.3d 1202, 1205-06 (9th Cir.1999) (holding that there is no “fundamental right to drive” and affirming dismissal of complaint based on state's refusal to renew citizen's driver's license); Hallstrom v. City of Garden City, 991 F.2d 1473, 1477 (9th Cir.1993) (finding no constitutional violation where valid Idaho law required driver's license, and plaintiff was detained for not having one). Without vehicle licenses, Dean is denied only “a single mode of transportation-in a car driven by himself,” see Miller, 176 F.3d at 1204, and this does not impermissibly burden his right to travel.
Matthew v. Honish, 233 F. App'x 563, 564 (7th Cir. 2007).
As a result of these cases and a number of others just like them if you are charged with driving without a license and try to defend on the notion that the state law requiring the license violates the federal Constitution you will lose. It is very well established law that the state may impose a driver's license requirement. If the sovereign citizen arguments, which have been around for at least 30 years, actually worked the Supreme Court would have by now struck down driver's license laws and states would not longer have them. So don't buy into to what you see on web sites or wherever else that peddle the long discredited notiong that states cannot require a driver's license.
If you strongly believe that licenses should not be required then lobby your state legislature to repeal the law requiring them. But good luck that unless you can propose some other useful method for ensuring that people who drive can do safely, track the violations of those who can't drive safely, and keep that latter group off the road.