It's not just the saddle height. The top tube might be too long, requiring a shorter stem that could affect handling.

My '89 steel road bike is technically the right size but due to an old neck injury it felt too stretched out on the original stem. I replaced it with a shorter stem which is much more comfortable. But I needed a couple of cautious rides to adapt to the slightly different handling, especially on fast downhills and turns. It's not twitchy, just more responsive. Feels fine now.

And if you install tires that are larger than were common way back when, it'll make the bike taller. One of my hybrid bikes is technically the right size, but was originally spec'd with 700x32 tires. After I put on 700x42 tires the bike felt considerably taller. The bike still fits. But I need to be more cautious setting a foot down at stops. Next time I build up a city bike from an old steel diamond frame bike, I'll go a size smaller to accommodate my preference for bigger tires.