It depends on the video. In the past, YouTube made videos available for embedding - copyright or not. I would go back to YouTube and see what the current limitations on that video are. IF it is still available for embedding, I would try to contact the owner and offer payment, explaining the issue. Often there is a minor licensing fee - but sometimes it is no longer available for embedding. YouTube has made many changes over the years, so it primarily depends on verifying the current arrangement for those actual videos.

If you are in violation of their current terms, then yes, they are enforcing their rights. Please note that I am not a lawyer, this is not legal advice.

YT has the copyright terms for the videos, uploaders can choose the different terms for their videos,
I think it's OK to embed them on your website if the video uploader choosed the CC or some other terms like this,

Creative Commons (CC) does not always mean that it is OK to use content. CC offers various levels of copyright restrictions. The video might be listed as CC3 or CC3.4 - there are over a dozen different CC terms. Before you use it, it can save you trouble by finding out what those different things mean.

Verify that the copyright notice is valid. There are those who phish that "look" legit but are not.

Meanwhile, contact content holder (as suggested above) to learn THEIR terms of use.

OR do it the hard way with options available under Fair Use (if USA) and CREATE a snippet that is 3-10 SECONDS of the video for purposes of scholarly report.

OR, provide a LINK to the vid and hope your user returns.

OR figure out a different way to do what you do without using third party sites or copyrighted material.

NOTE: Over the last year many things have changed and what once worked no longer will. The EU is about to pass revised copyright stuff that will set the web on fire for those who have used other people's content on their sites.

The handwriting has been on the wall for a few years... this year and next will see it implemented.