Find/Replace Font is replacing extra fonts & not embedding?

I'm sure there's a setting to solve my problem, but I can't find it. Here's the problem:

I have a InDesign file that, over the last 6 months or so of copy-paste-generations, had collected a dozen random paragraph styles from content being pasted in. I removed all those paragraph styles in one fell swoop, but I must have done something wrong. Now, when I try to print the PDF through an online printing service, the automated service tells me that Lato Regular/Bold aren't embedded in the PDF. (Which is weird, because it's always embedded it before.)

I've tried setting the  subset font % to 0%, but it didn't work. I tried switching fonts (to Proxima, Myriad, Calibri, etc) via the Find/Replace Font menu. However, when I switch the "Lato Regular", "Lato Bold" switches to the new font type with the bold version without prompting. (I mean, it's kinda neat to be able to say change all Lato styles to their corresponding Proxima styles so that bold, italics, and all the other weights switch automatically, but given that it won't embed in the document, it's really frustrating!)  Also, when I switch the font to anything else, the new font won't embed either. Even system fonts like Calibri!

I've gone into the Basic Paragraph settings and tried "Reset to base" and "Break Link to Style" options. Nothing works. I've updated, restarted, cleared all the App data and settings (even though I hate doing that), uninstalled and reinstalled the entire lato family, and outlining the text so there were not fonts to embed. (Not a long-term solution.) So what am I missing?