Scotland has a registry of all baby names registered in the country since 1974, available here. I dowloaded these files, and grouped the names from 1974-1979 and the names from 2015-2019. I then did the simplest thing you could possibly do to check this hypothesis, and ordered the names by the change in the ratio of the number of times they were used for boys vs. girls. I excluded any names that were used less than 5 times in total in the 5 year periods I was looking at.
My conclusion is that the original assertion was basically correct - there are quite a few names that have changed from predominantly male to predominantly female (Kennedy, Sydney, Morgan). However, there are some examples of names going the other way - in particular Regan and Corrie. 'Jan' also appears like an example, but I think this is just cultural - 'Jan' is a fairly common name for Eastern European/Scandinavian boys, and a pretty old-fashioned name for Anglo-Saxon girls. Lee does seem to have gone from being pretty gender neutral to predominantly male (and also to have dropped in popularity a lot).
Code I used to do this is available on Github https://github.com/johnfaben/baby_names, although you'd need to download the data yourself, as I'm not sure I'm actually allowed to share it unprocessed. Lists of the top 10 names that changed below.
Names that are now more commonly used for girls