A barbarian's Rage ends early

if your turn ends and you haven't attacked a hostile creature since your last turn or taken damage since then.

Players of barbarians, including those in my own play group, have expressed frustration with this limitation in any 'large map' combat, where enemies are spread out and movement between them takes more than one turn. This frustration has generated many questions on this site about whether rage can be maintained RAW by such techniques as; obtaining a mount (even if not culturally appropriate), keeping a bag of small animals to kill, attacking oneself, poking oneself with a needle as an object interaction, setting oneself on fire, and attacking creatures that might not be present.

While my game play itself is not always serious, the themes, motifs, and settings are. It would hurt suspension of disbelief for barbarians to adopt such 'silly' techniques as those above, even if they were functional RAW.

On the other hand, I would be fine with permitting an approach that was influenced by real world warrior techniques that did not involve attacking an opponent or taking damage every six seconds, such as a Maori haka peruperu or a Cheyanne Hotamétaneo'o staking himself with a dog rope, which would not RAW maintain rage.

I am considering granting the following ability to certain barbarians in my campaign, specifically those who can articulate an in-game cultural practice used to maintain rage:

Focused Rage You can also maintain your rage by spending your action to do so. This cannot extend the duration of the rage beyond its limit of one minute.

Presumably the designers of the game had a reason to limit barbarian rage in the way that they did, and also to provide Persistent Rage as a high-level feature. Designer reasons are off-topic, and I do not intend to ask about their reasoning here.

However, as part of a house rules question, I think it is fair to ask what exploits exist for a barbarian who is allowed to maintain rage without attacking or taking damage?

and then also

does my proposal to require the barbarian to spend its action to maintain rage sufficiently limit these potential exploits?