The Tzitz Eliezer has a T’shuvah (I want to say at the beginning of Chelek 22 but I’m not positive) where he discusses if shoes count as a Begged. He doesn’t specifically discuss in regards to Shabbos, but that’s definitely a place to start. If I recall correctly, he would probably agree with your husband. However, the Minhag very clearly is to be Mechabed Shabbos with special shoes-especially someone who wears sneakers/loafers etc. during the week.
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