There are times when a stiffer brush might be needed, just not on upholstery. I generally use a stiff brush for things like stairs or car mats. A stiff brush may do permanent damage to many upholstery fabrics and is not worth the risk IMHO. Also, I'd just X out that spotting brush. I haven't purchased one in over 30 years and I don't plan on ever purchasing one. It is really intended for spotting procedures on carpet and is pretty much useless for upholstery. It is a very stiff brush used to tap over a spotting towel to work a spotter into carpets. It is NOT intended for scrubbing! It will do permanent damage even to carpet if used as a scrub brush.
Now, with the money I just saved you get a good sea sponge for under 20 bucks U.S. If you have an Interlink Supply or can get from them I highly recommend their sponge in place of a sea sponge. It works even better and will last a LOT longer. Simply rinse it out thoroughly after every use. Here it is if you can order it I recommend you do so. It is for use with most fabric shampoos. I cannot stress enough just how much I like this sponge and it's about 1/2 the price of a real sea sponge.
https://interlinksupply.com/index.php?item_num=1697-6615 If you do upholstery or really fine fabrics you NEED this sponge or a good sea sponge!!!!!!!!!!!!!!!! Just my REAL opinion. I don't mean to be too dogmatic.
Btw, sea sponges work FANTASTIC too. They just don't hold up well over time like the Interlink sponge. You WILL NOT likely find a similar sponge locally. These are synthetic sponges made especially for shampooing upholstery. They do a really good job of mimicking a real sea sponge.
When you are using a horsehair brush it is to get your upholstery prespray worked INTO the fibers. It is NOT to SCRUB the filth OUT OF the fibers. Once your prespray has penetrated the rest of the work is simply rinsing it out with your machine. Getting too aggressive with a brush can do permanent damage. Btw, even horsehair can become REALLY stiff. I occasionally take mine out and actually use some regular human hair conditioner on it to soften the bristles and rinse it out just like washing your hair. It restores it just like conditioning your hair if you have any left! Those horse hairs can get stiff enough to damage a fine fabric. Don't ask me how I know.