MAJOR UPDATE ON REBLOG/LIKE ISSUE!
I’ve been contacted by @staff (THANK YOU!), and I now have what appears to be an answer to the problem. As I suspected, it’s something on both end: the tumblr owner, and the audience. Here’s what it boils down to:
IF you are a VIP USER (no idea how this happens, or how you know if you are one), then BLOGS THAT ARE MARKED AS EXPLICIT (presumably done in your own settings, but it may be the case that your blog may be marked as explicit by Tumblr even if you haven’t marked it as such—this is just speculation) you will only be able to interact with VIP user Tumblrs in a limited capacity, meaning you will not be able to reblog their posts.
This is an option that can be turned off by Tumblr, so I have asked them to do so on my blog, meaning that you will not have that issue with my tumblr anymore (yay!). But, for those of you who, for example, have reported you’re unable to reblog @npr, that problem will likely continue. Your blog will have to be not marked as explicit before you can interact with a VIP user the same way you can with a non-VIP user. Obviously is this is important to your Tumblr, you’re going to have to create a side blog to reblog VIP user content, it looks like.
Thanks to those who filled out my survey! Somehow whether the blog you’re operating is marked NSFW was not one of the questions I included. My bad! Nevertheless, I do know there are some that reported problems that don’t appear to be something different, so if you’re not able to reblog my posts by tomorrow or so (whenever the switch gets flipped on my blog), let me know!