@jay718 "this is now fixed!"
Whilst LogMeIn devs have improved the behaviour of the Quick Search (or "Site Search" as they seem to refer to it, from within the app itself), making it at least just about usable, it is still a good way off what I'd call fixed.
Problems I'm still seeing:
1. The UI of the "Site Search" window always appears at the bottom of my screen and is obviously not interacting with the macOS GUI as search results appear off screen and are unreachable, in the UI.
2. Selecting to copy a username or password takes normally longer than 10 seconds before it actually does the action. So, again, the UI of the "Site Search" does not interact with the OS well.
3. Selecting to open a site, from the "Site Search" UI causes Safari to open, when Chrome is set as my default Web browser. This one has already been confirmed, by LogMeIn support as "by design". However, this is, more accurately described as "by design error", as the UI is clearly ignoring what is supposed to be a system wide user preference that has been set.
So, credit to LogMeIn for getting this part of the app to the point where it is just about worth considering to use it. Let's hope they are able to improve it to make it actually usable and not take a couple of years, this time.