Yes, a PhD in CS would open more doors, but there’s no reputable PhD program that remotely focuses on “creating software and analysis tools”, which isn’t research. BTW, a good PhD program in machine learning is harder to be admitted even with publications and research experiences which OP doesn’t have (some of these programs have less than 1% acceptance rates). What OP really wants to do is in education (even though I’m not sure a PhD is necessary). Some CS skills are certainly helpful in OP’s stated objective as they are in so many other fields.