No issues for the overflow, millions of cars made that way.
Environmentalists may not like though
If you fill the radiator to the very top,,, It will drip (usually after you shut it down) and basically level itself off to a point where it stops doing it. And you are good to go.
An overflow however isn't a bad idea and having a closed cooling system can have some advantages. There are many aftermarket and factory style tanks that should fit your taste.
A neat way to do the 67-72 trucks (and I've been thinking about doing it myself) is to take another washer bottle and bracket that is used for the washer/wiper system and mount one on the other fender (passenger side) for an overflow. You just have to change the cap to a dual nipple design. One for the radiator hose and the other is your drain or vent.
Doing it this way on a 67-72 truck makes it appear like it's a factory setup...