Use single method to open/close flyout

[ericholscher]

Fixes #541

[agjohnson]

Lit provides a native way to update dependent attributes/variables on reactive property changes. We should use that pattern instead of relying on manually calling methods like this. With willUpdate(), the current code would be fine

https://lit.dev/docs/components/lifecycle/#willupdate

[humitos]

Take a look at the tests I wrote and try to write a test case for this case. These cases where we need to interact with the elements to do QA are pretty hard to notice after deploy otherwise.

[ericholscher]

Cool... I'm a little lost in general with all of this, so might take a while to get back to this.