Plants do benefit fish considerably, but to grow them you need to give them a fair chance. I now have 10 densely planted tanks in which the plants are thriving with minimum effort. I only have "ordinary" fluorescent lighting and no CO2 injection, so don't believe those who insist you must have enhanced lighting and extra CO2 to grow plants successfully. What plants need first and foremost is a decent substrate. Again, no need to buy ridiculously overpriced substrate additives. Mine is very simple, an inch of garden soil on the bottom, topped off by another inch of sand or gravel. Don't be afraid of soil. It stays under the top layer and won't make your tank or water dirty. I've found plants really thrive in this. However, it's best to restrict yourself to "easy" plants. Those which I've found to do best are Hygrophila polysperma and H. difformis, Rotala rotundifolia, Sagittaria subulata (doing much better than Vallisneria) with Echinodorus tenellus as a dwarf plant for the foreground. Cryptocoryne, and Java Fern or Anubias attached to bogwood, are OK, but they grow pretty slowly so I doubt if they are very useful in terms of improving water quality.
If you just put an inch or two of sand or gravel in the bottom of your tank and stick your plants into it then you are greatly reducing your chance of success. In nature, such substrates are invariably devoid of plants and one can hardly expect plants to succeed in such unsuitable conditions.
I can't do better than the following quotes from Diana Walstad's "Ecology of the planted aquarium".
"Essential to my natural aquarium is moderate lighting, a substrate enriched with ordinary soil, and well-adapted plants.
Hobbyists can protect fish from toxins by hard work, e.g., frequent water changes, gravel vacuuming, and enhanced filtration. However, given a chance, plants can purify the water naturally and effortlessly for the aquarium hobbyist. In my opinion, the ability of plants to purify aquarium water and protect fish has been woefully underestimated.
Using soil in aquariums is a strong ideological barrier for many aquarium hobbyists. Here, I mean soils that ordinary gardeners grow plants in— garden soil (i.e., topsoil) or potting soil.
Tanks with a soil layer and healthy plants will remove ammonia naturally, so bio-filters are unnecessary and possibly counter-productive.
Artificial fertilization with CO2, trace elements, and macronutrients is unnecessary if the tank contains a fertile substrate, the fish are fed well, and nutrients are not removed by over-zealous tank cleaning."
I suggest one must decide from the beginning whether to have a planted aquarium or not. I suspect most problems occur where people don't consider the requirements of the plants before they decide how to set up their tank, then try to grow them as an "added extra".