Best to call a mason. They may be able to add on to what you have without a complete demo.

Composite deck on your home will look cheesy.

Does your entrance have a canopy to protect the door and guests?

There are precast steps with railings available.

Talk with a composite deck craftsman and the Tech Support people at the manufacturer. If you treat this as a deck installation, I think you'll find well developed solutions for a brick facade.

Keep doing your research until you know exactly what will work. Then pick the right person to do the job. Supervise at the start and fire anyone not following the process you want.

I have no idea what your house looks like so cheesy who knows .I think a much wider stoop is a great idea but we do need to see your house from furhther back. The brick steps can be removed DIY with a bug sledge hammer and some muscle and IMO thta would be what you need to do then build anew stoop and stairs out of some material . Please post the pic of the house in acomment here I do not think just because the house is brick the steps need to be brick.

A new deck will require framing support and it doesn't look like you have enough height between the existing brick landing and your homes finish floor.

Here is the outside photo of my house. Hope this helps @Patricia

Not sure the bricks are in great shape (they look like they need to be leveled?).....but, if a mason could fix them and expand them a bit on each side you might be able to cover them with something like TechnoBloc. We did that a few years back and have been very pleased. Railings would also be a nice touch.

Using composite decking allows you to widen the porch along the house with a ledger board on each side of the current porch, footings and two posts. Steps can be as wide as the porch if you want.

You should be able to build over your existing steps. Your problems might be:

1) you'll need posts to support the ends of the new steps, which might have some movement over time.

2) if your planned new steps are a mismatch with what's already there. (I doubt this will be a problem, if the new steps are deeper).

Trex does not look cheesy. We have some composite as stairs for our 100 year old house. Sure, if you can look, you can see the difference between the composite stairs and the wood porch, but you have to be looking.

I have done this before. The existing steps were not level, nor flat.

I built a frame from wood, including hand rails and support for a canopy.

Just covering the steps.-