Try loosening the the grease nipple where it goes into the oiler bolt ,they have a long thread, then see if you can get any grease in and or if any escapes from the threads around the oiler bolt. This might be enough to clear a blocked oiler.
Exactly what grease gun are you using, I'm afraid it does have to have something of the he-man about it and if the nipple attachment end is cheap or plasticky then It just won't be powerful enough or secured enough. I have used hypoid 80/90 gear oil in a gun before to help free blocked oilways, you'd be surprised how little it takes for them to become blocked with hardened dry grease. You only need to introduce a little, probably under 5 mls and that usually helps move things along!
The lower exposed part you asked about, I suppose you mean the lower area of the main spring where you can see the pin? And possibly below that where the small rebound spring is? Yes keep it clean! In fact aim to keep all your suspension spotlessly clean. Grease is constantly finding its way out ,both passively and when in use. More so when you have just greased them! The airflow around these parts tends to spread it all over the suspension and hub and this carries grit and muck with it and soon compromises the efficiency of the suspension, and steering and you have described beautifully how dry or dirty suspension starts to be compromised with for example, the self centering ability becoming less efficient and the ride will become much harsher too!. You'll start getting kick back through the steering wheel and the front end will get very crash/bang.
On the bright side, this is very common as a problem. Owners often believe that as the grease won't go in it must be full and not need any so accept that all is well when in fact it's not. when sorted you will be surprised how much grease you will put in and over 500-1000 miles most will find its way out! Hence the dirty suspension. I might be a tad obsessive but I clean my suspension after most trips, two at mast with a visual after every journey. Makes cleaning easier though! I think you are in for a real treat when your's is all cleaned and lubed up as it really will transform the car.
