Is it considered filial piety to one's mother to reduce visits to her and suffice with phone calls, or to visit her at relatives' homes to avoid embarrassment due to her preventing her maid from entering upon him, to the exclusion of other family members, knowing that the mother is pleased with this arrangement?

What your mother advised regarding the maid's hijab and her not mixing with men is correct, because she is required to wear the hijab like other women. And what your mother did in preventing the maid from entering upon you is a correct action and not intended to embarrass you. So, you should not abandon visiting her at her home. Rather, you must be patient and explain the matter to others, that the prohibition was due to your advice regarding the obligation of the maid's hijab.