There are many online services that analyze social media data and then
display aggregate results...
Those online services may be doing the analysis and aggregation under the site TOS; or they may be doing that with permission of specific and individual license agreement that they have negotiated; or they may be doing that without explicit permission and under what they feel is Fair Use, and we don't know that.
From what I understand this use case would need to fall under "fair
use" doctrine, but it seems quite vague.
It's vague because YouTube's overall TOS - written by YouTube's legal team - needs to be broad to take into account many different instances of what they see could be valid usage as well as cases that they feel infringe and could pursue legal action against. They're trying to explain Fair Use in the TOS, but don't want to make it sound like they condone all usage while making sure they make it understood that Fair Use is a legal doctrine. The TOS is a legal document that both YouTube and users abide by and would be used in a (civil) court case to interpret the contractual arrangement and the possible limits of how Fair Use applies.
So if I understand correctly displaying some captions metrics would be
fair use if I don't include any quotes? What about a summary? It could
be considered the “heart” of the work and therefore use would be
What is really "Fair Use" and not would be ultimately decided by a court. And it's up to YouTube to initiate legal action against you and other people that YouTube considers infringers. They may or may not take action, depending on if they feel they could win; or at least discourage your use by costing you money to defend yourself, if they feel they couldn't win on the merits of the usage in terms of Fair Use. Or they may just ignore your usage; that's also an option.
You can look at Fair Use case law, but even a case very close to your type of usage might be determined differently by a court.
You could try contacting YouTube for clarification for what you want to do, but due to the volume of email they get, you may not get a response. It's your choice to aggregate and analyze with your interpretation of the TOS and Fair Use and without direct permission from YouTube.