Simpler ways to do this all appear to require buying a third-party app or uploading your PDF to a potentially less trustworthy platform. So, this seemingly over-complicated method is the simplest I've yet found. Please tell me if you know an easier one!
(If you're a programmer, please help make flattening a PDF a standard export option... it boggles my mind that it's not.)
A Google Drive open_in_new account
iOS default apps: Files open_in_new and Safari open_in_new
Save it to the Files app.
…such as a Word doc, first convert it to PDF.
Convert to PDF with the Google Drive app (easier)
Note: If you don't already have the app, here's where to get it open_in_new.
In Google Drive app, beside the file, select more_horiz › Print › iOS_Share › Save to Files › Save.
Now, the file is a PDF in the Files app.
Caution: If the non-PDF file is there too, be careful not to confuse them later on.
Convert to PDF without the Google Drive app
In the Safari app, sign into drive.google.com open_in_new.
On the left side of the address bar, select menu › more_horiz › Request Desktop Website.
Note: Your Drive should now look as it does on a computer. If it doesn’t, refreshing the page may help.
Upload the file to your Google Drive.
Select New › File upload › Choose Files, then find, select, and Open your file.
Note: This may take a moment. If you don't see the file you uploaded, select My Drive, then you should see it under Files.
Next to the name of the file, select More_vert › Open with › Open in new tab.
Select print, which will open the PDF in a new window.
Note: If this option disappears, you may have to tap the screen to see it again.
In the bottom Safari menu, select iOS_Share › Save to Files › Save.
Now, the file is a PDF in the Files app.
Caution: If the non-PDF file is there too, be careful not to confuse them later on.
Open the Files app.
Find and open the PDF.
Select AutoFill Form, either from the pop-up at the top, or from the dots & pencil icon in the lower right.
Note: These options may take a moment to appear.
Now you should see blue fields where you can add text.
Tip: If any blue fields are missing, or if the AutoFill option didn’t work, you can add and format text wherever you like.
Select the pen tip icon › add › Add Text › Aa (to format). Select then drag the text box anywhere.
Select the pen tip icon › add › Add Signature, then either add a signature or select one you already made.
Select and drag to move or resize your signature.
When you’re done filling out the form, it’s important to “flatten” it, so that other people cannot easily change your answers.
In the Safari app, sign into drive.google.com open_in_new.
Note: This must be done from Safari in order to work. Wild, I know.
On the left side of the address bar, select menu › more_horiz › Request Desktop Website.
Notes:
Your Drive should now look as it does on a computer. If it doesn’t, refreshing the page may help.
If you have previously requested desktop website, then it might still look as it should.
Upload your completed PDF form to your Google Drive.
Select New › File upload › Choose Files, then find, select, and Open your file.
Note: This may take a moment. If you don't see the file you uploaded, select My Drive, then you should see it under Files.
Double-tap your uploaded PDF to open it.
Select print, which will open the PDF in a new window.
Note: If this option disappears, you may have to tap the screen to see it again.
In the bottom Safari menu, select iOS_Share › Print › iOS_Share › Save to Files › Save.