How can I bring my stained carpet back to life without using a carpet cleaner ?
You can try using a carpet shampooer using only water.
A carpet cleaner is your best bet. It will scrub out the dirt, suck up all the dirty water, and you can also finish with a rinse. It may take a few times of cleaning with the cleaning solution to get all the stains up and rinsing a few times to get out as much dirt/dirty water as possible. You can rent one if you don't have one. It's not a fun job but maybe a faster and more efficient way to get the stains out. You will be pleased with the results.
Consider painting it! Look on line for how to stain or paint carpets.