Swetter

Well, JohnnyD, it's hard to say what's causing your problem.  No, you don't need to have the blade installed to run the engine.  But the weight of the blade does tend to provide a bit of centripetal force which will better support the starting effort.  How old is your mower?  The reason I ask is that Sears doesn't list any parts for it that I can find which, in my experience, suggests it's been out of their inventory for a very long time.
Have someone check for the presence of spark at the spark plug.  If you know how to do that safely, without causing a fire due to the gasoline fumes that are likely to be present, you may want to try that yourself.  There should be a nice rich spark when you crank it over.   Run a Google search with "Briggs 5.5 GTS, OHV" and click on the sixth entry on the page.  Is that  your engine?  If so, you should be able to find a manual for it on-line or through Chilton.
With the spark plug in place and with the choke fully set, if you pull it through a couple of times and then remove the spark plug is the spark plug noticeably wet.  If so, and if you've got spark, and assuming you have some resistance to the pull you place on the starter rope when it's pulled there's a pretty good chance you have a running engine.
When gasoline is left in an engine for long periods of time, it tends to turn to varnish.  There are some small passages in the fuel system (deep inside the carburetor) that can get varnished shut and it's not always easy to get those cleaned out.  If you used only gasoline to clean the carb. perhaps you would want to consider a quality carburetor cleaning agent (one that you can soak the carb. in for a period of time) that is available through your local auto parts store.  Read and follow directions - that stuff is extremely toxic.
Safety is the primary consideration when working on your engine.  Don't work on any engine unless you absolutely certain you fully understand the consequences of your actions.
One last point.  Before you put a lot of money into the mower, check to make certain you can still get blades for it.  I recently had to go through a great deal of work to convert an old Sears Craftsman 12HP riding tractor mower in order to get a blade to work because nobody, not Sears or anyone else, could provide blades for it.