i think his hand should be dead if he lets significant action happen afterwards?
I think this is correct as the overriding fundamental principle... did the player "let" significant action happen before stopping the action. To retain the right to act, a player must stop the action by calling "time" (or an equivalent word). Failure to stop the action [in a timely manner] may cause you to lose the right to act.
Personally, I see a couple of issues here. First, I would refer to the definition of "substantial action" in the TDA rules as a starting point. If substantial action has occurred, then by default I would rule against the missed player. However, I think there can be exceptions to this.
There will be times when technically, significant action has occurred, but I would feel comfortable ruling that the action be backed-up. For example, if it is clear that the player did not deliberately delay calling attention to the first out-of-turn action once it has occurred, and attempted to stop the action
right away once it had occurred (e.g. despite subsequent insta-calls or insta-folds), then I think it is fair to back the action up. Conversely, there will be times when technically, significant action has not yet occurred, but I would not back up the action. For example, if only one player acted, and the dealer then tapped & burned & dealt out board cards and yet the player did not attempt to stop the dealer's action, then I would rule that he has lost the right the act. I know that some people will disagree with me here, and strictly apply the "substantial action" standard, but I do think some common sense is warranted. The question, in my mind, is did the player let play continue after the out-of-turn action or not?
Second, what does it mean by "losing the right to act"? If the missed player was contemplating calling a bet, I would rule the hand dead. If, however, all action was checked around on that particular betting round, I would not kill the hand but deem that he has checked and he loses the right to bet, but his hand is still live. I really don't like killing hands as a matter of course, so I think this is a fair compromise.
In the original example, I think it really depends on when seat 3 brought attention to the out-of-turn actiom of seat 4, and whether he delayed in doing so. I don't think seat 3 has any obligation to say anything while seat 4 is merely "reaching for chips", but once seat 4 actually acts out of turn by announcing raise or by placing chips forward, then seat 3 has to attempt to stop the action right then and there. If he did, but seat 5 acted so quickly that he couldn't be stopped in time, then fine, I would still back up the action. However, if there was any delay after seat 4 had raised, because e.g. seat 3 wants to see what would happen next, then seat 3's hand would be dead as soon as seat 5 mucks.