Both are needed by the plant. You won't do anything to improve that plant but you have a pup coming out the side.
Wait until it is about 2/3 the size of the mother before removing. Mum may put out a few more pups before she is done.

That is not a crispy "central brown stalk", it's the dead flower head of the plant. Leave it be and as Sunshine pointed out there's an offset forming so everything's good.

Gonzer I don't think that flower ever met its full potential.
Mine usually last nearly a year until they look like that.

Broms are so hardy that for many it takes a while to dry out because their leaves can hold moisture very well. By the same token, if dried out, it can take a while to 'remoisture' them. A mistake people often make is to water them in their pot and often this leads to root rot. I take them out of the pot, clean the roots and stick them in a pail of rainwater for a day. Totally submerged. A good soaking. Then after their soaking, I put them in a shaded breezy area and watch them. Sometimes they need another soaking if they have harder leaves. Many times, I pull them from their pot and I take the garden hose sprayer and I clean them very deeply. This dislodges crud that can cause root rot especially in Neo's. Remember that most broms can grow without soil so let those roots have some air when they are under stress and you will find the pups just come along and surprise you! You can fix this plant no problem!

No amount of soaking is going to rejuvinate that dead flower.
Galiana you can cut the dead flower off to make the plant look better.