Normally some hot water and detergent and a scrub brush with some elbow grease will bring a linen strop to like new condition no matter how dirty it is.

Get some bick leather cleaner. It's quality stuff, does a great job and is very gentle on leather.

I wouldn't be comparing a saddle to a strop. The leather is totally different. Saddle leather is really tough stuff that takes pretty much any abuse and laughs at it. A strop is way more delicate.