What advantages did they have that their adversaries lacked? What little I can find on them says they were very well trained, was there any sort of horsemanship culture in Finland at the time? What made Finnish, not Swedish, horseman preferred?
I think the reason you can't find much about them is that their fame is largely a product of contemporary Swedish/Finnish nationalism rather than historically founded. I have read a lot about the Swedish Army and never seen any mention of Hakkapeliittas preforming better or worse than a cavalry regiment raised in Sweden proper.
I'm going to go out on a limb here and say that the Hakkapelitta were not invaluable at all to the Swedish army. At least no more so than any other regiment. If I were to rank which regiments were most crucial to the Swedish army under Gustavus Adolphus I would say the artillery and the Guard as these were his only true professional regiments, the infantry and cavalry being semi-professional soldiers under the allottment system.
I did a litte digging through orders of battle for a few major battles in the war. Att Rain there was only a single Hakkapeliitta squardron present (Stålhandske Cuirassiers) out of 27, the vast majority of the cavalry was german. At Breitenfeld they were 12 out of 127 companies of horse on the field. All in all, I can't find any indication that they were considered any more effective than a regular cavalry regiment and certainly not invaluable.
I would however call upon /u/vonadler and /u/mp96 to see if either has more to contribute, both have written great posts about related topics in the past.