This verse is easier to understand if it is combined with the previous verse, Matthew 23:6. The active verb for both verses is in the prior verse ("they prefer"). The "first couches" and "first seats" of the previous verse from a word the means "first" or "highest" plays against the "rabbi" here that means "great." The "prefer" of the previous verse, which can mean "greet with affection" play against the "greetings," used here though a different word.
The verb that means "show sign of affection" from Matthew 23:6 plays with the noun that means "embrace" here. There are also two different words used in these verses that both mean "assembly" or "place of assembly," one here referring to a marketplace and other in the prior verse, meaning a meeting or meeting place.
Since "rabbi" is a Hebrew term, it should not be translated in a translation from the Greek. So this is correct.