dnd 5e – Does healing an unconscious, dying player character restore only up to 1 hp unless they have been stabilised?

No such rule exists.

There is simply no rule that says healing a creature with 0 hit points only counts as 1 hit point. The rules do say this:

The best way to save a creature with 0 hit points is to heal it.

A creature receives full healing from any healing spell while unconscious and at 0 hit points, and they immediately become conscious again:

This unconsciousness ends if you regain any hit points.

These quotes are from the rules for Dropping to 0 Hit Points.

Further, the rules for healing state:

When a creature receives healing of any kind, hit points regained are added to its current hit points.

If your current hit points are 0, and you receive 13 points from a cure wounds, the rules say you have 13 hit points.

A Grave Cleric feature explicitly confirms that you roll dice for healing a creature at 0 hit points.

The Grave Cleric’s Circle of Mortality feature states:

When you would normally roll one or more dice to restore hit points with a spell to a creature at 0 hit points, you instead use the highest number possible for each die.

If it were the case that healing an unconscious creature always only restored one hit point, this feature would be entirely useless: it would never trigger because you would never be rolling one or more dice to restore hit points.