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 action each turn if they want, forever. (Does unlimited Wild Shape make Circle of the Moon archdruids virtually unkillable?)

You asked if this is Rules As Intended. The only source I know of for this is the Sage Advice site; I searched for 'primal champion' (https://www.sageadvice.eu/search_gcse/?q=primal%20champion) and found no hits. I also tried searching for Jeremy Crawford's tweets and got no hits.

Your DM has the ability to interpret these rules if they wish. Your DM might rule that this ability should end early if you get knocked unconscious, if your DM feels that this is what's best for their game. We can't tell you how your DM should rule.

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 action each turn if they want, forever. (Does unlimited Wild Shape make Circle of the Moon archdruids virtually unkillable?)

You asked if this is Rules As Intended. The only source I know of for this is the twitter aggregator site: www.sageadvice.eu; I searched for 'primal champion'and found no hits.

I also tried searching directly for Jeremy Crawford's tweets and got no hits.

Your DM has the ability to interpret these rules if they wish. Your DM might rule that this ability should end early if you get knocked unconscious, if your DM feels that this is what's best for their game.

We can't tell you how your DM should rule.