"White" has been used as a common term for a race longer than anyone living has been alive, so, no, I don't think anyone alive could miss the days when "white" wasn't a race.

People can, I suppose, miss the days when "white", as a term for a race, had almost exclusivey positive connotations when used by anyone with any influence in society and pine for the days when white privilege was so strong as to be virtually unquestioned, but you shouldn't mistake missing that for missing "white" not being a race.

In a history of American cities, I read that the Polish, Irish, Italian, German, etc. identities shifted to 'white' around the 1930s or 1940s (my memory is a little hazy).

Its true that before then, Polish, Irish, Italian (German not so much, AFAIK, but possibly) were ethnic identities often viewed as distinct from -- and frequently discriminated against by -- the dominant white-identifying group (in the same what whites of Hispanic origin have continued to be since), but white racial identity existed before then.