'Effective aperture' is a problematic concept.  A very long (high gain) Yagi is another example where the concept seems to fuddle practical comprehension. Personally, I just try to ignore it.

When an antenna is dielectrically loaded, it's usable bandwidth also tends to become more narrow.

Antennas frequency band shifts by sqrt Er if you don't change the antenna size, or size changes by same sqrt Er.

bandwidth generally reduces proportional to the sqrt Er

waves generated in the dielectric tend to make it less efficient (though you may not notice it unless you're really trying to squeeze every dB of gain out of a tiny antenna).

effective aperture (directivity/gain) would reduce alot if you have a high gain antenna and embedded it in dielectric, and hence it's size shrinks alot, but if it's a small antenna like a monopole, you see a smaller reduced effective aperture change when you load it with dielectric.

A note on directivity versus gain. Directivity is just related to the pattern shape (i.e. narrow beam is high directivity, wide beam is low directivity). Gain is identical to the directivity if the antenna and it's combining network is totally lossless.

Directivity formula = 41,000/[(Beamwidth Az)*(Beamwidth El)]
A common number for Gain replaces the 41,000 by the number 30,000 to account for a small amount of loss and the loss due to the VSWR of the antenna. example: A standard waveguide antenna with 10 degree beamwidths has gain = 30,000/(10*10)= 300,
Usually 10*log(Gain) is specified by manufacturers, hence this gain would be 24.8 dBi(L).
kch

The efficiency of your antenna will be very dependant on dielectric material. As You probably know most of dielectric materials are much more lossy then air. Also they are trying to keep electric field inside. So, if You want minimize the size of antenna with dielectric load, be carefull. You will achive much smaller dimensions and very good matching, but emmited power might be "slightly" different then you expect.