The notion that you get prion diseases like BSE only by consumption of brain is not correct. Any part of the infected animal that contains nervous system tissue or lymphatic tissue can be infectious, although I guess this is dose-related.
Quote from OIE, "BSE. General disease information sheet" ():
The risk linked with consumption is addressed by the requirement for
routine removal of all visible nervous and lymphatic tissue (Specified
Risk Materials (SRMs)) from carcasses during the processing of
cattle as well as the removal of any suspect animals from the human
Appropriate handling of nervous and lymphatic tissue tissues from
BSE-suspect animals in abattoirs and laboratories is indicated to
avoid accidental exposure (ocular or oronasal). Milk and milk products
are considered safe.
The OIE is the animal equivalent of the WHO.