Just a nitpick - for most artillery, the time of flight for a shell over those sorts of ranges is going to be more like 30 seconds. And that's if the gun is already loaded and aimed at the target, if they have to do that (and the fire control calculations) as well then 2 minutes is a more realistic figure and would be pretty good performance. Heavier guns take longer to load and aim as well.
I can see a role for this sort of ship in areas with bad terrain, where waterways are practically the only form of transport - parts of South America or Africa spring to mind, maybe Southeast Asia as well in places. Getting conventional artillery into those places would be nigh-on impossible, but floating a ship like this up and mooring it in or near the operational area could be handy. Still, that's treating it more as an artillery raft than a monitor.
Exactly, which is why its stretching the convention of what a Monitor ship is to its absolute limit. Even then its mostly a naming issue not a class issue.
Such a ship would only be a Monitor in name, bearing almost no similarity to its historical counterpart and being more of a artillery raft.
And my concept for it would probably work best with two small independent crafts with a combined command where when one ship is ready to fire the other ship is not. That would cut down on the reaction time.
Automatic artillery with the minimum possible human components so much so that the ground crew can automatically enter coordinates that are then transmitted to the nearest ship and to the fire control computer/staff bypassing the normal chain. That would also cut back on the time.
Either way, you could get a artillery platform with a response time well below 1 minute for anything inside a 20 or so km radius from the platform itself.
If you keep at least one weapon loaded at all times, which has its risks.
If you allow for direct target coordinates by the ground team to the fire team, potentially automatically and using a form of GPS. This would probably be best done in some form of automatic system where the position of every soldier is known within a few meters and all they have to direct is "fire at 300m in that direction".
It would require quite an accurate command and coordination system for the platforms, and redundancies that allow severeal ships to have weapons ready in any low intensity conflicts.
If it is done extremely well and the risks are understood, you could have a fireplatform with the ability to land something equivelant to 200mm artillery shell under a minute in any target in the field.
Combine 3-5 of these near any coastline with the appropriate force required and 1-2 in active duty and 2-4 in reserve and once a ground team needs fire support you could fire the 1-2 on active first while the 2-4 on reserve prepare their system and within roughly 5-10 minutes of calling for it the ground team could have something with the firepower of a Iowa BB at their disposal anywhere within 20-30km from the coastline.
If said platforms/monitors are designed solely as ground artillery support in difficult terrains and low intensity conflicts you could have a sizable cheap artillery presence anywhere in the world. This would probably be a massive thing for something like a low intensity pacification in a place like the Niger Delta or Somalia. You could build, man and fire 10 of these at the cost of much less than 1 missile frigate.
Anyway, its still not a Monitor ship on its own but its an interesting concept, "it has its faults" is a way to put it lightly, but other than that I really dont see a place for the Modern Monitor, that is unless we have some kind of star trek like shields in the near future.
EDIT:
Thought about that targetting issue quickly and the best way I can see of is a hand held device with a quickly adjustable range setting and a laser pointer and a compass.
A GPS picks up the location of the device.
A Compass the direction
And the laser either looks at the range to what is pointed at (by calculating time of laser return)
or failing that last one due to target being invisible or in cover
You can set the range manually.
The computer then calculates if any known devices or units are within range of the target radius and fires.
Only problem with this is that at a range of 20km from the ship, the shells have a very large area of impact so at longer rangees its not exactly useful for hitting a specific house.