How can I update this fireplace?

Paint the screen! I personally love the multi brick. The mantel does look dated. If you intend to paint you should be able to cover up whatever damage is done to where you removed the sides. Add a frame around the screening and purchase two posts to add from the top of the step to the edges of the newly cut mantel. You can add trim to the fronts of the border around the screen by cutting on post in half and gluing one half to each side of the frame you have built. I would not just PAINT the brick. Whitewash it instead. A totally white painted wall will look dated. Water down your paint and apply with a rag you daub on. Try two or three shades very similar for compliment. Practice on a piece of cardboard to find a color combo you like. Clean the top of the step and leave brick unpainted so it is not monochromatic. It will add an accent to the wall.

Paint the fireplace surround with a high temp. black paint and paint the brick also. Both will give the area a modern and clean look. Depending on the color you choose you might want to paint or stain the mantel too. Would give you some more storage space if you put floor to ceiling bookcases of each side to the fireplace too. That would help break up the entire brick wall look. If you built them yourself you could even make them so they fit around the heath that protrudes out.

Leave the brick, it's pretty. Change the mantle and update the brass.

you might consider putting book shelves on both sides and like Naomie mentioned new mantle and new fireplace doors to match the light fixture in the room.

If the brick "sides" go up beyond this room, through the attic and outside, for example, then removing the sides will not be cost effective.

Yes, bookcases on each side, would work nicely.

Personally, I don't like smoke stains on my brick and painting it doesn't change the maintenance hassles of brick.

Replacing the brass doors is going to be pricey and I would spend less of that money on this:

I would veneer right over the brick and the cut-down mantle with a glazed or a porcelain bodied tile, that requires no grout joints. That would be called "gauged" tile.

No cuts would be required if you ran the tile to overlap where your book cases would be installed on top of it.

If the brick is pretty regular, then you might not even have to use thinset and might be able to get away with mastic. But my guess is thinset.

We're talking $1-2 a square foot for the tile and $30 in thinset or mastic. No tile saw rental. Crown molding at the top, butts into the bookcases, if the tile courses don't line up with the ceiling.

You'll be doing the prep work no matter which you choose . + tile work @ 6 man hours. And clean up, no matter which you choose.

Then I would re-do the floor with a tile that compliments the FP tile for Phase II. Avoid shiny glazed tiles for floors (slip and fall) porcelain tiles are fine.

Before you do anything, get a chimney sweep in there, as they are messy. But a chimney fire can be much worse than messy.

Just think before you paint the bricks for future selling of this house because once painted it is not coming off without damage to the bricks and some buyers may not be interested and you can only repaint it? You decide it is your home.

They always paint the bricks white in home shows and then use ship lap to finish the walls

I agree the brass looks dated. You could paint it in a high- temp black. Just be sure to sand lightly, but enough to give the paint something to grab on to. The book shelves are a nice idea. If you can find shelves with cupboards underneath, so much the better. Keep in mind if they are not built in they will need to be fastened to the wall to keep them from tipping.

Here is what we did to our ugly brick fireplace. Maybe it will give you some ideas.

https://www.hometalk.com/diy/living-room/fireplaces-mantels/the-saga-of-the-truly-ugly-fireplace-33847745