Hey 👋🏼
I’ve taken a quick look – the code that’s being added for the videos is precisely the same, whether the video plays or not. I suspect a broken tag on your site or a plugin is interfering – I see, in the JavaScript console, an error being generated by that subscription pop-up. That may, or may not, be related here.
In fact, those videos that don’t work… the links underneath them don’t either, so something is definitely very wrong.
I put your page code through a validator (validator.w3.org) and, as I suspected, there are numerous formatting errors – end terminators missing, etc. All sorted.
For a start, I’d try temporarily disabling that subscription pop-up to see if that cures it. If not, I’d go through disabling all your plugins to see if any are at fault? What’s changed since this started happening? Has your theme updated, or will it just be plugins changing? If disabling plugins doesn’t work, I’d consider (temporarily) switching to another theme too to see if that helps at all.
Sorry I can’t be more precise – there isn’t a cause jumping out that would allow me to pin it down any more exactly.