Can a restrictive covenant deny a land owner the right to use a road
on their own property?
A restrictive covenant could impose such a limitation, although usually a court would disfavor that interpretation of a covenant unless no other reasonable interpretation is possible from the language of the covenant.
This is basically because it would be lawful (at common law anyway) to convey the land in its entirety to the covenant beneficiary, so it is lawful to convey a lesser interest to that person.
It isn't inherently unconscionable unless it makes the property subject to the covenant landlocked with no access to a public road, in which case some mandatory doctrines regarding a right of access would probably apply.
Often this would be done to create a private drive across the "front lot" (which has direct access to a public road by other means) when the "back lot" of a parcel was conveyed by the owner of the combined lot to the new owner of the back lot.
A covenant might be chosen instead of an outright conveyance of the dogleg of road because it retains mineral rights in the servient estate owner, because it clearly allocates property tax payment duties to the underlying owner as part of a larger economic deal, and/or because it could make a transaction possible that would otherwise be prohibited by subdivision requirements applicable in that locality. The restriction might also reduce the servient estate owner's liability for accidents arising from use of the covenant access easement, since it would ordinarily vest all maintenance and use regulation in the dominant estate owner.
Some other common easements that deny the servient estate owner all use of land for a period of time are easements to allow staging of road construction activities, and easements incident to mineral rights in states where destruction of the surface is allowed if it is later restored when the minerals are extracted (such as Wyoming).