shaft drive has been around for a long time.
https://oldbike.wordpress.com/1898-pope-mfg-columbia-chainless-shaft-drive/however it is expensive to make, inefficient, inflexible, heavy and (a bit like a belt) feels a bit weird when you pedal hard. Without knowing anything about it you can quite easily conclude that if it were 'just better' then everyone would use one already.
Nonetheless the perceived disadvantages of a chain are such that folk are prepared to overlook the issues with shaft drive and such bikes still exist. Typically modern shaft drives use a special frame (of course) and at the rear hub a kit of parts that bolts on; I have seen several recently which use various forms of Nexus IGH. Some of them look quite nicely made, with helically cut gears which run both smoothly and quietly. However as with a belt drive, it is a double (or even triple) whammy, efficiency-wise; you have whatever losses are in the shaft drive plus whatever losses are in the IGH, and because you don't have any choice about the overall gearing, you can't even choose you make the most-used gear ratio the most efficient one, either.
If you have a shaft drive coupled to a Nexus 7 then you might be doubling the losses on average; for your effort to get to the rear wheel it has to go through four sets of gears when using the middle three gear ratios in the hub.
Mechanically speaking, one of the attractions of the shaft drive is that the mechanism is enclosed and maintenance free. However in many forms the bevel gears are not perfectly protected, and they need regular greasing. Needless to say they usually get totally neglected (which is the whole point of the thing really) and should they need repair then spare parts are not to be found easily.
From the POV of the shaft itself, it is probably best if the gears at the ends of the shaft are made close to the diameter of the shaft, in order that the shaft is suitably stiff in torsion. It wouldn't be too difficult to do this by having a large diameter hollow shaft and large diameter gears. However the choice that is usually made is to have a relatively skinny shaft and rather small diameter gears (thus making the 'sprocket' gear about half the normal diameter). 'The meshing efficiency is OK even with the small gears because the tooth pitch on the gears is short' some folk would argue. And they would probably be right; there are only small gains to be had in terms of gear efficiency by making them larger; the efficiency is never going to be that great anyway. However this ignores two things
a) that the shaft drive is liable to be all squishy as the shaft twists up and
b) that the thrust loads on gears are larger for any given torque whenever the gears are made smaller, and these loads are reacted by the bearings in the hub. These loads are often two or three times larger than those that might be seen in a chain drive. [ Its a bit like choosing to run 24/12 chainring/sprocket rather than 48/24 in that respect.]
So no surprise then that the RH bearing in shaft-drive equipped nexus hubs is a common point of failure; these bearings are a weak spot anyway and need no extra encouragement before they are liable to fail.
One aspect of shaft drive bikes is easier than you might expect, and that is rear wheel removal. It is about as fiddly as with an exposed chain I suppose; the biggest impediment is often the rear mudguard. Often there is a cover over the 'sprocket' gear which has to come off and then its just a matter of disconnecting the gear cable gubbins, undoing the track nuts and sliding the wheel out backwards. Simply replacing the wheel is also not difficult; usually the axle is located on the RHS e.g. by the flange on the RH locknut so that you can't get the wheel in the wrong place. If a Nexus 3 is used then a single NTW on the LHS is enough and this simplifies things on the RHS of the hub.
However when setting the thing up in the first place the axle needs to be shimmed and there is similar awkwardness if the position of the axle is wrong lengthwise in the dropouts; there is only one 'right place' for the gears to sit and if you get that wrong then the result can be as rough as you can imagine, worse than the most knackered chain and sprocket imaginable.
So the adverts saying 'makes climbing hills easy' in 1898 were basically a lie; gear ratio for gear ratio even a knackered and neglected chain is liable to be far better in terms of efficiency. With shaft drive (moreso even than with belt drives) it gives no choice over the direct drive ratio; the required variations in gears simply don't exist, and if they did, it wouldn't be easy to change them for different ones; e.g. you would need some means of varying the length of the shaft.
So if you want a clean and reliable transmission on a utility bike, a chain inside a chaincase is probably a better solution. Chains inside chaincases often last a good fraction of the life of the bike (about 20000 miles is not unusual), with minimal attention, and are the most efficient transmission possible. It is difficult to do better than that.....?
cheers