Stephen Carlson wrote:With the critical text, with καθαρίζων, the antecedent of this participle is the subject of the λέγει (Jesus) at the beginning of v.18. It is probably to be understood as an editorial comment on the part of the narrator that Jesus cleansed (i.e. declared clean) all foods, although subsequent Christian history (e.g. Peter in Acts) suggests that this was not fully understood at the time.
A little Relevance Theory on the Greek text might put Peter and Jesus in a much better light. The question in Mark 7 was whether a piece of bread or a fig became unclean by being touched by unwashed hands. Jesus declared them all clean. He rejected a secondary-level uncleanness of unwashed hands. See Yair Furstenberg. Eating in a State of Purity during the Tannaitic Period: Tractate Teharot and its Historical and Cultural Contexts.
Peter got that right, then and in Acts.
So the Alexandrian text has the Gospel writer explaining Jesus' halakic implications, pointing out that Jesus is the one καθαρίζων all the foods under suspicion of transferred uncleanness.
The Byzantinan text καθαρίζον refers back to παν το εξωθεν. That text becomes part of the quotation of Jesus. It may be seen as a 'more difficult reading' because the logic may be blurred or confusing. The 'external food (touched by unwashed hands)' does not defile a person and the example of this case cleanses other such cases. That 'food' cleanses all the other 'foods' by not making a person unclean. No handwashing is necessary and no secondary uncleanness is meaningful.
The compact nature of the Byzantine reading is more complicated than the more direct and simple reading of the Alexandrian reading.
However, the halakhic background is the same to either reading.