Tested on Chrome 77 and Firefox 69

Left clicking the bar to gain focus shows the caret blink once before the focus is lost. Sometimes it works fine if you let the mouse stay perfectly still as you click, but most of the time the focus is lost.
Focus is also lost when you move the cursor away from the bar.

Focus is NOT lost when there is a suggestion open.

This affects both the main bar on top and the small one in the search results page.

This is a very annoying problem. Happens to me a lot of times.

I wonder if this is because the mega menu has things that react to being hovered over.

Yes it is. The search bar loses focus as soon as the Mega Menu stops being hovered.

Usually when I click the search bar I move the mouse down to start typing. Because the mouse crosses the mega menu options, the search bar loses focus and I type in vain. In the past I even tried to type in the search bar several times not understanding why it wouldn't work.

I usually click and move the cursor away a little bit so it doesn't cover the text.

I've messed around with it and I found out that

1. it loses focus when the cursor leaves the mega menu area (either just the navbar when it is closed, or the whole menu when it is expanded).
2. moving the cursor over the mega menu and quickly clicking the search bar makes the search bar lose focus almost immediately
3. it doesn't matter if the suggestions are shown, if your mouse moves over the mega menu it loses focus too

This applies to both the search bar on top and the one in the results page