What I was told and read was not that the risk was due to NSAIDs and platelets, but due to the risk that NSAIDs can cause stomach bleeding, particularly in higher doses than OTC and over several months. Ibuprofen is $1 for 40 pills at my local dollar store. You'd be surprised how a little bit can be effective with joint issues.FWIW - I'm going through a situation where my elbow is inflamed, and the rheumatologist extracted about 8 ml of blood from the swelled area. He suggested that I take an anti-inflammatory. I told him that I take warfarin and NSAIDs are probably not an option.
He told me that the safest NSAID (it has the least impact on platelets) is Celebrex. He ordered a prescription for a few days of Celebrex. Unfortunately, insurance formularies don't cover this under insurance (it's too similar to many OTC NSAIDs), and I didn't want to (and couldn't) pay almost $34 for just a few pills. I was also worried that, if I had blood in my elbow, NSAIDs may make the situation somewhat worse.
The point here -- Celebrex may be an option for inflammation -- if you can afford it.