For some people, it's not a matter of taste and it's not a matter of getting over it. Call it weakness if you want, call it mental scarring; either way movies should be entertaining, not traumatic. I can't speak to dogs dying, but flippantly watching "Last Night in Soho" with somebody who really didn't need to see that convinced me to start checking the IMDB parental guide before settling on anything.
The structure of the referenced website (long list of yes/no categories with explanation) seems a bit of a bad fit compared to just enumerating potentially problematic features. I guess it enables categorical searching, but it seems pretty bleak to browse through a filter like this.