After some tinkering, I ended up figuring it out!
I at least wanted to post a follow-up for any future users experiencing similar issues.
So what I did to fix this was go into Advanced>Blog Post Item Code Injection and make the following changes to the code.
<style>
.entry-header {display: contents;}
.entry-footer {display: contents;}
</style>
It took a lot of time in Inspect Element to trace why the header image was being forced into each blog post and what class the image was directly affecting but I got it.
In a previous attempt, I had the display set to hidden instead and while that did hide the image I used as a header, it also hid metadata such as likes, tags, author, and title, so I don't recommend that unless that is your intention.
Just as a quick recap, I am using Squarespace 7.0 with the Bradford template. My Original code was fine (although admittedly suboptimal and amateur.)
