In general, I don't spend much time worrying about "neglectedness" in the ITN framework. I think that is because while many important problems are not neglected in an absolute sense (e.g. climate change), it is still the case that some solutions will be much more effective than others within each problem area. Therefore, one may have a large impact on less neglected problems by simply focusing on more effective solutions.
Are emergencies different from non-emergencies? A new paper ( https://link.springer.com/article/10.1007/s11098-020-01566-0 ) argues that the obligation of saving a drowning child is different from the obligation of donating to effective charities in order to save a life. They claim that in emergencies where we can directly intervene to save a life, we are obliged as participants in an informal insurance scheme in society to intervene even at great cost to ourselves. Through this model they aim to explain the "common sense" moral intuition that it is worse to ignore a drowning child than to not donate $3000 to the Against Malaria Foundation. Overall an interesting read that may be of interest to EA-aligned folks.