We snake through P traps almost on a daily basis going through the trap inlet NEVER though the CO plug under the traps
What happens when your retrieving the cable and there is a big clump of hair on the end of the cable ? It will not come though the small opening most P traps have
Removing the trap causes another problem as 99.% of the traps are brass screwed onto on a galvanized nipple and once the trap is removed to snake the line the galvanized threads are gone from electrolytic action so now we are replacing the nipple and trap
So we compromise by removing the waste fitting above the trap and then snake the lines though the trap inlet
Yesterday I was on a job and and used a trap leader on my 1/2" cable to allow me to easily pass through a 3 " cast iron P trap for a floor drain
With the right sewer /drain cleaning cables or even using a General Ram Jet most P traps are very easy to pass a cable thought it is what is behind the taps that can be a challenge such as short turn fittings or back to back installations
Clog chopper easily makes turns though 3" openings
https://drainbrain.com/en/generals-clogchopper-tieger-tail/
If the stoppage is not roots or a rag etc then the Ram jet will work as I have 6 of them ( 3 are for sale)
http://old.drainbrain.com/why/waterramsuccessstories.html
Also I use a Water Cannon 2 HP electric Water Jetter 2 GPM with a 1/8th stainless steel hose (made by Mustang) hose and it fys thought reaps and ells . The rubber hoses also work but have a lot more friction
Want to buy a General Super Vee