Why does the cold water only trickle in while filling the tub?

Possibly some sediment clogging up a filter.

Try turning off the cold water where it's hooked up to the washer, and remove the hose. Have a bucket handy...

There may be screens on the hose ends and one of them is likely filled with sand, iron, or whatever. There might also be a screen on the washer inlet, so look there, too.

When you reattach, first connect the hose to the water faucet and crank the valve open to see if it flows like a hose should, into the bucket. If not...you might need a plumber.

If the screens are clean, water flows fine from the hose, then there's possibly a stuck or clogged solenoid in the washer.