How do you add more soil to a mulch covered area?

It would be okay to just add soil on top of the mulch. The resulting mix of soil and cedar mulch will resist erosion. Fir bark is a good type of mulch that will decompose to create soil. You just add a little each year to maintain the look of the bark chunks.

You are right the mulch does disintegrate over time plus it blows away. Depending on how much you want to build the area up you can either add more top soil and then mulch which is the cheapest way or you can simply add more mulch. Either way you do not need to remove what you have there now.

You will need to remove the mulch that is down and then add the soil and then put the much back on top.

if there is an arboratum in your area go visit there and see what plants and flowers grow under their trees. It would be better if the rubble were removed, but if you haven't the energy or the funds then try growing some rhododendrons, philodendrons and ivy. If you can get it leveled and cleaned out then you could make a nice pathway and put in a bench and then go enjoy the comfort. Get a Burpee or other seed catalog and see what grows in shady areas and what worls well in the soil you have. Maybe the grass dies because it lacks nutrients or that specific grass that you had or tried to grow isn't the kind for shady areas. If it's covered with weeds then surely some kind of grass will grow there.

Add new compost over top of the mulch. Turn all that over. Rake level. Add new mulch.

You can use a hoe and move it from one side to another and add it like that