I wouldn't trust a fly line to 20 lb running line in the winter...not so much for the fish but for when you wrap your tip around a rock. If you're fishing a floating line in summer, I'd think it would be fine.
I personally use 36 40 47 and 50 test
When new and perfect 20 will do fine but running line gets alot of abuse and gets kinks and weak spots as you use it and if your Fishin deep your asking for trouble even when new.
In short: you bet, but it's on the small side. 30# might be a better all around. 20# is often the breaking strength of fly lines up to about 7 weight. However, the other posts about rock dings are accurate. The exterior coating on fly lines helps to protect the inner core of a fly line (the 'strength' of the fly ine) from getting knicked and losing strength, to me one of the big differences between mono running lines and fly line running lines for steelhead use.
With all of these comments, though, mono running line has been used effectively for steelhead fishing for years and years.
In general 20lbs is fine, considering all the stretch of the line and leader it's hard for a fish to put enough pressure on it to break and your tippet should always be light enough to break first. Of course if your tip is wrapped around a rock it can be messy but that's rare.
It also depends on the diameter of the line. Airflo's 20lbs Ridge is .036 which is easy to hang onto and plenty strong but many other 20lbs lines are much thinner. Also Cortland coatings are not as slick as Rio, SA or Airflo either so it might not be as good a deal as it sounds in the long run. The other problem is the word is Skagit/Sauk and most or all of the other Puget Sound rivers are closing at the end of Jan. this year so you won't get a chance to use it much.
For me the wost thing about the smaller diameter running line is when you are fighting a good fish it will pull down into the spool. Then it will not come off the spool evenly and you find you break the next fish off. The larger the diameter the less this will happen.
I dont think that you will break the 20 lb as your tippett for steelhead is usually in the 10 lb range. I am switching all my running line (exvept my trout spey) to 30. I find that it tangles less when shooting line. I am still using the rio slick shooter/ compact skagit/ t14 for my king setup.
It wouldn't bother me as far as breakage concern, I use 25# amnesia and love it...I know it's 25% more than the 20# your speaking of but even 20# I'd be confident in not breaking. ON a different concern...as far as mono, I'm actually thinking about going with the 30# while shooting a 480grn head. I hoping to get a little more stability in the roll-out with a heavier running line. The 25# may be a touch too light for this application, though while fishing I have every confidence that it won't break. The beauty about using mono is the value. Sure it's got it's quirks, but its cheap and highly functional. Changing it before every big trip and or every couple of years, it won't break the bank and you'll always be fishing. Go to Ebay and get a few spools of amesia and you'll be set for years for under $4 per spool....is the cortland that good of a deal?