What version of Acrobat Pro are you using? I can't seem to reproduce the behavior you describe using Acrobat Pro DC. In answer to your question "How can I be confident my pdfs are accessible?" - you would first need a better checker than the one built into Acrobat Pro. The free PAC 2 from Accessibility for All is a much better choice. In addition, you would need to carefully inspect the document for the many accessibility issues that cannot be detected by any automated checker. The free Matterhorn Protocol provides conformance criteria for PDF Universal Accessibility. Of the 136 failure conditions, 47 require human inspection and judgement.
I also have no such issues (Acrobat DC Pro) -- I just tested. I never used that wizard in Acrobat Pro 8, 9, X, or XI, though. The "Make Accessible" process is not really intended as a check and report process. It's a wizard designed to lead a less experienced user through the various steps in making a PDF accessible.
Open the Acrobat Accessibility Full Check dialog box. Ensure that the Checking Options area reads, "(32 of 32 in all categories)." Perhaps it is not doing all the checks.
I echo "a_C_student" in strongly recommending use of the free PDF Accessibility Checker (PAC). I use the Acrobat checker as a first-cut as I work. I use PAC as the "standard." PAC also comes with some features
- Check of all 136 failure PDF/UA-1 conditions
- A report that shows the location of an error on its page
- a screenreader display that shows how screen reader software will read the PDF (it can help you find errors)
- and more...
I add a screen capture of the concise PAC summary report as the last page of a PDF when I submit it for final review and upload to "wherever."
I also highly recommend Kenny Moore's TaggedPDF.com site, Sec508 PDF Help Center, and related (YouTube) videos of how to use Acrobat to fix varied accessibility failures found by PAC.