Earlier quoted context omitted.
"admits" seems to be a frequent weasel word these days. You identify the issues with someone's story that you don't like and then you make a story about how they 'admit' those things. God help anyone who actually 'admits' anything these days as to some people that must be instantly discrediting.
"Claim" is my favorite of these words. Main example: "Elected official claims nothing bad was done" It has a gentle enough negative connotation to allow you to massage the emotions of your reader, while still being a technically accurate statement. Only people with something to hide will "claim" or "deny" something. It's the same idea where you're talking about something and your audience can feel a "big old but" com…
Think of an insurance claim and/or civil claim. In response to a civil complaint initiating a civil lawsuit, the defendant must answer the complaint by “admitting” or “denying” each allegation.
And “allege” is the entire basis of US criminal law...as in innocent until proven guilty. As in the defendant allegedly committed the crime until and unless convicted.
These are all legal terms of art.