Unfortunately, nothing like this exists in Lansweeper.
You would need something like PingInfoView, MultiPing, WhatsUpGold, SmokePing etc, there are plenty out there. Just depends on the amount of detail you are after.
I would look at something like MultiPing (Windows if you're after a basic up / down etc). If you're talking external accessible sites etc, then check out https://uptimerobot.com/