For a large and/or high speed fan, there are several mechanical disadvantages, for example
The outer ring is highly stressed and therefore heavy. It may also need a heavy containment system to avoid collateral damage if it breaks. For example in a large turbofan jet engine, a mechanical failure caused by a crack near the hub of the fan usually only releases one fan blade. A crack in the outer rim would be likely to destroy the complete fan.
The blades fixed to the rim are compressed radially, instead of being in tension when fixed to the hub. That means they have to be designed to resist buckling which might compromise their aerodynamic shape.
With no central hub, you need a bearing the would support the entire rim of the fan, which would be large (and complex) compared with supporting a central shaft.
A fan adds energy to the air mainly by increasing its tangential (whirling) velocity, not its axial velocity. (If the objective is the increase the air pressure, the tangential velocity is reduced by static vanes behind the fan and converted into a pressure increase). Therefore even if the hub blocks say 10% of the total area of the fan, additional airflow through that 10% would create a much smaller increase in useful work than 10% because the tangential velocity of the air is lowest at the hub. (Note, even if the hub radius is as large as 1/3 of the outer radius, the hub only blocks 1/9 of the total area, not 1/3 of it.)
So the advantages of "increasing the effective area" are less than the disadvantages. For example the thickest part of the blades would have to be at the outer radius, where they would obstruct the air flow in the most efficient part of the annulus, not at the hub where they only affect least efficient part of the annulus area.