I always wait until the pups are atleast 50% of more the size of teh mother plant before removing them. Survival rate is much better and plants are healthier

It kind of depends on what type of Neoregelia you have whether or not it will look better if you remove the pups. Some stoloniferous ones like 'Fireball' etc. look better as big clumps, and the stolons hold them far enough away from the mother plant that they don't suffer from crowding. Most, though, will start to look crowded. This doesn't really hurt the plant any, and you can leave it that way if you want, it's more of an aesthetic thing since the form of the rosette is an important feature on a lot of Neos. Also, you are likely to get more pups if you remove them than if you don't, but if you're not interested in propagating the plant then that won't matter.

As hotdiggety says, if you do decide to remove them, 1/2 the size of the mother is a good time to do it. I don't have any trouble removing and starting Guzmania pups either where I live, but I've heard many in less tropical climates say they are very touchy so maybe best to wait until they're almost mature size. Or you can leave the clump together and just put the whole thing in a bigger pot, as you suggest. The mother will start getting brown leaves sooner or later, but just remove them as they appear, and in time the whole plant will be gone, but the pups will be mature by then, and will more than fill in the gap.

When you remove an old flower head it's best to cut the stalk as far down as you can, just because an old stump looks ugly and serves no purpose to the plant.

Thank you for your advice. By looking at photos online, it seems I may have one of these two cultivars: Rose Blush or Carolinea var. tricolor, though I tend to lean more toward Rose Blush based on the purple color in the middle. Maybe I can take a photo and upload it.