I've already upvoted your request and this feature would be nice.
Also the category "Feature requests" is correct.
After more than 5 weeks waiting, without reaction of a PGC team member, moderator or 'heavy user', I'll want to confirm, that's not possible to search for coordinates in Live Map at the moment. This feature would be nice for 'sometime mobile website cachers' like me.
Here is my workaround, that I'm using during business meetings in other cities.
I use simply the 'map compare'-Tool with a 15 km centered coordinate.
15 km is the 'stressless' range for a folding-bike, in my view...
Feel free to change the filter / map center coordinates:
Map Compare Tool - Filter for specific map center coordinates
Of course, this is not a replacement for your suggested feature within Live Map, but a possible first step workaround and a first solving orientated answer in this thread...