I would add vinegar to the water..it helps with getting soap out. Is it the carpet or the pad underneath? The pads seem to break down long before the carpet is worn out.

Sounds like the pad is destroyed.

Not the pad. Do not hear anything when walking. When sitting on it, the fibers are somewhat stiff. I feel that it probably has been cleaned by the homeowner either improperly or many times and it has left it how it is.

I wonder if it's possible to get water hot enough to clean synthetic carpet that would melt the fibers? That won't wash out.

I would suspect shampoo and/or essential oils. My son bought a house where the woman who lived there before him was an essential oils dealer and HEAVY user. The carpet felt strange and cleaning it again and again didn't help. Most carpet is made of synthetic fibers and are resistant to water and soils, but oils in the air or in the shampoo can interact with the synthetic fibers and change the texture of them, directly or indirectly ( making the surface sticky or gummy and causing dust, dirt and other things to stick to the fibers that used to repel them). You can try cleaning only with water or steam, but if that doesn't work, it may be best to bite the bullet and consider replacement. Sorry.