How to export PDF with trim size as the page size, still showing bleed

That is a dreadful workflow and it's amazing they have gotten away with for so long!

Illustrator is not, and I repeat IS NOT, an all purpose PDF editor. A pdf from InDesign has a different file structure to that of an Illustrator PDF. Only PDFs created in Illustrator with Preserve Editing Capabilities should be opened in Illustrator. 

If they need to edit the trim size or set art/trim/crop/bleed/ use Acrobat Professional and Set Page Boxes

You can export a PDF without bleed and crop marks from InDesign, but you can't use it for print when bleed is needed.

You can also export it with bleed (crop marks are often not needed) and then use an Acrobat Preflight Fixup to crop it to Trim size- Set crop box to trim size.

This cropped PDF still contains the original bleed and can be used for printing*, although there is a good chance the cropped PDF will confuse the printer.

The Cropped PDF can be un-cropped in Acrobat, or placed into InDesign, where the bleed will re-appear.

Set the page size properly in InDesign and for the love of god find another printer. Editing PDFs in Illustrator is an absolutely horrible thing to do.