Over the years the accumulation of soap scum and other debris (lint for one) collects on the internal piping walls thus reducing the capacity.
Snaking will only punch a small opening in these types of drainage lines.
Picture putting your finger in a pile of sand as soon as you take it out the hole closes NOW put a garden hose into same pile a WHOOSH it is flushed away
Now this gives you two options.
1- Have a plumber water jet your lines to a like new condition restoring FULL flow as the piping was designed Or
2- Control the flow by having the discharge of the washing machine go into an indirect waste pipe into a laundry tub
By using this laundry tub it will hold the volume of the machine and give your drain lines a “controlled flow” and thus not over load your system.
Also if you are currently using the normal 2″ stand pipe of washing machine waste removal and this line does get blocked up you will have a flood as this 2″ pipe cannot hold the volume of a machine discharging.
You should also consider the use of low suds detergent for laundry.
Have fun