How to compare a magnetic loop antenna to a conventional antenna
It's pretty common to use a ferrite core for signals around 1 MHz. Consider the standard ferrite rod coil in a regular AM receiver: -

Picture from here.
It uses a ferrite rod because it improves the signal amplitude received by concentrating the transmitted magnetic field part of the prevailing signal through the core and therefore inducing more signal voltage into the coils. Given that the coil is also an inductor, that property can be used to give a tuning capability when used with a capacitor.
The longer the rod the more field it will concentrate through the coil and the bigger the signal.
It's still pretty good if really compacted. For instance, when using near-field power transfer, even a mere hint of ferrite material in the right place can significantly improve the received power.
The Fair-rite material 61 you linked is very good in this area and I have used it several times for improving power transfer between coils: -


Does the core introduce noise beyond what is described with the
complex and real permeability?

Given that this technique is used in radio receivers (that deal with very low incident powers from distant transmitting antennas), I don't see a problem and I've personally never seen one. I've even pushed the envelope with type 61 material and used it in a data transformer for bit rates of circa 600 Mbps and it proved a useful and beneficial addition.
Given that the core will cannot be a continuous loop (like a toroid) in this type of application, there is no danger of core saturation or significant non-linearities.