For instance, when English, Yiddish, and Russian were written down, they borrowed pre-existing scripts of their religions-- Latin, Hebrew, and Cyrillic, respectively. Are there any examples of cultures in, oh, the last 500 years, creating their own alphabets instead of borrowing existing ones?
I am not a historian and I have never posted here before. I'm not sure if this is helpful and I am not an expert. I do however have first hand knowledge of the answer I will provide in the form of learning the language and living in the country of origin.
I believe that the Korean written alphabet (hangul) fits your description. It was created just under 600 years ago (1443) and as far as I can tell was created from scratch using the positions of the mouth. Again, I am not a linguist and I hope anyone who has a greater knowledge of the subject will provide further information.
Here is the wikipedia entry on hangul.
Again, I am sorry if I don't meet the requirements stated by the subreddit to answer your question, but I thought I might as well throw it out there.
As already mentioned, there was Hangeul, Inuktitut syllabary, and Cherokee Syllabary. There is also the Vai syllabarry in Liberia and Sierra Leone (invented around the 1830s); N'ko for Mandinga languages in Mali, Guinea, the Ivory Coast, and other parts of Western Africa (1949), Afaka for Ndyuka a creole in Suriname (1900s I think).
I guess you could include things like Bopomofo, but that's really just a different way of writing Chinese by syllables. Of course all of these were generally developed by people who knew of other scripts, even if they chose to make their own. Those in the Americas knew of Latin script, as did some in Africa. N'ko was apparently created based off the creator's understanding of Latin and Arab scripts.
I should say that knowing of other scripts is not that surprising. Writing seems to have been invented independently only a few time: In Sumeria and Mesoamerica for sure, probably China, and I've never been able to keep up with the debates about how much influence Sumerian writing had on Egyptian or Harappan writing. If someone more knowledgeable on those subjects wants to pop by, I'd be glad.
There were a few examples sprinkled through this post on aboriginal writing systems
The most prominent one in Canada is Inuktitut (ᐃᓄᒃᑎᑐᑦ) syllabics - here it is in action on the Nunavut government website
While this might not be in the spirit of your question, there are a few fantasy alphabets in very modern times that are significantly different from existing alphabets. Some examples of this are elvish, created by J.R.R. Tolkien, and the alphabet used in the dragon language in the video game "Skyrim." These don't have that much influence beyond those fantasy settings though, but they are alphabets, created in what is considered modern times, that aren't directly based on historical alphabets.