When I go to layout, and then to Page Elements, I'm given the option to "add a gadget" but not to "add a page element." The Add a Page Element button has never been displayed since I started the blog, so it's not something I broke myself.
Oh no!! How do I add a page element!!??
What everybody may not realise is that what Blogger used to call "widgets" are now called "gadgets", and adding a "widget" (weird term) / "page element" (boring term) is now called adding a "gadget" (descriptive term).
What we call the "Page Elements" wizard may be found as the "Layout" link in the dashboard. If you publish the blog using a dynamic template, layout customisations aren't supported - and you may not have "Settings - Layout".
Of course, there could still be cases where the "Add a Gadget" link doesn't appear, doesn't appear where you need it, or doesn't work for you. You still need to be aware of template restrictions, whether intentional or non-intentional.
When you look at the selections in "Add a Gadget", note the author names, carefully. The gadgets that are not attributed as "By Blogger" are provided by third parties, and should be selected with care.