I know everyone is having fun being outraged, but there would not be "cultural" targets on any US target list. Such sites are on a "no-strike list" and/or a "restricted target list" including all UNESCO sites and many others, incl mosques, schools, etc.
I did my PhD on target selection and vetting. It's a very laborious, mechanical process for fixed sites, & there is a huge constantly-refined no-strike list. Judge Advocate Generals are involved in all target lists. Dynamic targeting around restricted tgts is strictly controlled.
I would treat this the same as when POTUS has occasionally opined that we will do X, only to be told we can't / don't. In this case, I don't think he even meant material icons or whatever. I think he was talking about people.
There is an interesting case study back in 1990s with statues in Baghdad that were continually nominated to the target nomination board across different strikes in different years, never to be approved. Crossed Swords among them if I recall correctly.
