Definitely harder to find...
...these days, an MP3 player that can play video yet conversely doesn't have FM radio capability. Especially if you're looking for a model from a major brand (I don't doubt that there are examples from generic off-brands, especially with some of the schlock that emanates from China these days).
Archos makes a unit in their current lineup that might work:
However, I'm not sure who sells this model.
Amazon sells this Archos unit, but it's an older model:
The 3rdGen iPod nano would have been perfect, but new-in-box examples tend to be rather pricey nowadays (collector's items, I guess).
Refurbished examples are available, as are used ones. eBay seems to have a bunch of people selling new-old-stock as well for quite a bit cheaper...but of course it's eBay.