Yes, you are exactly right. Inflammation is part of the body's trying to heal. Basically what is happening is when we injure ourselves inflammatory hormones called prostaglandins dilate the blood vessels in the area to increase blood flow and thus oxygen and nutrients to the area to promote healing. When the blood vessels are dilated like this for a while though they become permeable so they leak fluids in to the surrounding tissues. This is the swelling that people experience.
Nonsteroidal anti-inflammatory drugs work by inhibiting these prostaglandins thereby constricting the blood vessels. This prevents the leakage of fluid, but also inhibits the healing process. In addition these drugs are known for causing heart attacks, strokes, chemical induced hepatitis, kidney failure, loss of vision and tinnitus because they can reduce or cut off the blood supply to organs and other tissues. And contrary to popular belief this does not require long term use or excessive doses. There were 2 dozen deaths from ibuprofen (Advil, Motrin, Nuprin, etc.) induced hepatitis during clinical trials. And I know 4 people that had their kidneys shut down after taking a single recommended dose of ibuprofen. The people most at risk are those with circulatory disorders such as those with Raynaud's disease or phenomena, congestive heart failure, diabetes, etc.
A couple of options to help with the pain and swelling without inhibiting healing are either using an arnica cream topically over the injured area, or internally licorice root and yucca root are both strong steroidal anti-inflammatories.
why would steroidal anti inflammatories reduce inflammation but not hurt the healing process, dont they essentially do the same thing? (reduce blood flow)
I guess the easiest way to explain this would be to compare it with phytoestrogens. Phytoestrogens do not contain actual estrogens. They just have weak estrogenic effects. When talking about anti-inflammatory plants the "steroidal" anti-inflammatories do not contain actual steroids. They contain sterols, which have mild steroidal properties and help support the adrenals that produce the body's steroids. Where they really differ from true steroids though is that like phytoestrogens that regulate estrogens, phytosterols regulate the inflammatory response. Even though they partially inhibit prostaglandin synthesis, which can constrict blood vessels, sterols also dilate blood vessels. This is one of the reasons sterol sources are used to lower blood pressure.