The part about the strainer being dirty, but not so dirty that it won't let oil go through.
That is not exactly saying the strainer is OK.
That strainer has a very fine mesh and if you can see anything at all on the strainer, its capacity to flow volume has been reduced.
I'm sure that the strainer was designed with an overabundance flow capacity and I don't know at what point (% of blockage) problems would surface related to reduced flow volume (which will directly affect the ability to build pressure).
To my recollection the suction strainer on those vintage units was the only filter present. At least the ones we had did not have an additional external filter.
You may want to try jdplayer's advice and try a clean out and new oil.
My take on the situation is: You have little risk in trying and if you find out that you need to go inside later, the new strainer will still be good.