will an old wheelset support 28mm (28c) 700c tires or will they "balloon" My bike is early 2000s and came with Shimano wh-r540 wheelset and 26mm tires

googling shows ETRTO: 622 x 13C (internal width?) and 19mm (external width?)

I've successfully used 25mm tires but considering going to 28mm since I believe there is clearance everywhere important for a tire that wide

However I am questioning if the rim is too narrow to properly support a 28mm and that it will "balloon" defeating the benefits of 28mm

But I've noticed new bikes that come with 28mm typically have a very wide rim to support the tire so the tire sits more square or inwards and not ballon out

ie. modern rim + 28mm tire profile =   /----\

ie. old 540 rim + 28mm tire profile =   \----/

does it matter? will it defeat the benefits of 28mm for handling?
Wider road rims have become more popular, in part because of the reasons you're talking about, but 28 is far from the reasonable maximum for a more classic 19-20mm road rim, and will be totally fine. Consider how many people race cross on 19-20mm clinchers with 32s and 35s. Where the actual line is of what's reasonable is a hotly debated topic that there are other questions about.