If you succeed on a Death Saving Throws 3 times, you don't recover any hitpoints. Instead, you become stable:
A stable creature doesn’t make death saving throws, even though it has 0 hit points, but it does remain unconscious.
The creature stops being stable, and must
start making death saving throws again, if it takes any
A stable ...
Three death save successes and you're stable but still unconscious
On three successful death saves a PC becomes 'stable'. A companion can also stabilize an unconscious PC with a successful medicine check, or by casting the cantrip spare the dying.
Check the rules for 'Stabilizing a creature' again:
A stable creature doesn't make death saving throws, ...
You are interpreting the rules correctly. A literal reading of these rules is that the paladin would become conscious at the start of each round unless they were actually killed.
This isn't necessarily unreasonable, and it's not necessarily the best level 20 capstone ability. Compare, for example, to level 20 moon druids, who can wild shape as a bonus ...
On reaching 0 HP
Once a character reaches 0 HP, they are unconscious, and starts rolling death saving throws on their turn. (See page 197 of the Player's Handbook.)
When rolling Death Saving Throws
When making death saving throws, there are 4 categories of results that we care about: rolling a 1 on the d20, rolling a total between 2 and 9, rolling a total ...