Anchor Points Malfunctioning
I regret to report that, after updating to Adobe Illustrator CC 2019 and upgrading my OS Mojave, the anchor points have become rather fickle, especially with the Direct Selection tool. The shape will look the same once drawn with the Pen Tool, and the anchor points will still be there, but they are invisible. They can still be tampered with, but after releasing the mouse they will disappear. The same issue applies to the rectangular grid tool. Anchor points can be edited and seen with the curvature tool; that is about the only way to do so without the anchor point disappearing after the mouse ceases hovering over the point.
Another issue regards the Selection Tool. When I click an object, a rectangular box of my selected layer color appears around said object, but no outline appears around the object. For example, if I draw a spiral and select it, a box will appear around the spiral, but the spiral itself will not be outlined in my layer color.
I would show you a screenshot of the issue, but Illustrator deselects before I can take a proper screenshot.
It feels like you had View > Show Edges accidentally turned off (Ctrl+H).
It can be useful sometimes, but yes, can lead to these exact problems you describe, if not intentional.