If the cold side is just clogged with grit, what is stopping you from just removing the used cartridge, cleaning it, then replacing it? I do agree for the long run, you may as well just replace the guts with something new, which will last for several more years.
make sure it's not the supply before you waste your time with hunting down a cartridge. Disconnect the supply and open supply valve into a bucket. if water good hold bucket under cold on faucet or disconnect supply from supply valve and reconnect to faucet hold end in bucket. have some one open faucet in middle mixed Hot/Cold position. Hot water might just back wash loose debris back out the cold side. put a rag against the spout for a little added pressure.