Yes, osho-sans from temples are not responsible for carrying dharma transmission or shiho, which is literary the same term as in soto.. inka is not precisely a document of dharma transmission. Anyway even oshos carry kind of shiho which they receive at their respective honzan, without it they cannot become jushoku-abbot.
Anywway what are they responsible for? Well the temple itself.. if it is not abandond temple in some deserted area, then present jushoku-abbot is responsible to pass the temple to the right person. Does not matter if it is son or not.. anyway still some public temples in Japan require single abbots,, i.e. not married.
So one abbot should pass temple to his student, and then it means he is responsible for his education, send young monk to rinzai uni, and for his training so sends him to proper sodo. And then there is a lineage of this particular priest, and it is a lineage of the temple itself. So in older temples often one can see pics or even panited portrets of the past masteres/priests of the temple.
Availability of shike to the public.. fortunately it is not that bad. Many are. I remeber just Engakuji's kancho, [kancho is a highest priestly rank] Adachi roshi who often could give public talks, and one could meet him just doing shopping. Yoshida kancho of Kenchoji used to be available. I think that many are easy going people, just it is matter of public - is it really interested? I do not think so..
so one story... just previous Kancho of Butsuji, one of the oldest and famous rinzai honzans, went really far, giving up completely the position of kancho shike etc.. Actually he was playmate of my elder brother so in a way it was immidiate info I got from the family. HE QUIT! out of the blue
pretty much shock for Japanese standards so I was really extremely interested what had happened. My bro just talod me that kancho and shike in one, went to med care school (?!) I was really puzzled. And I was fortunate to see him some years later talking on tv. So he went to med school, and became some sort of health care nurse, or whatever, anyway he worked with many different patients. And I saw him having a lecture with a wide audience of students from med uni. Future docs. I was very moved since he was talking from the depth of his heart and also of his human feeling. And really meant to help sick people in their suffering... it was his message to future docs about real pain and fear etc. and how to arise compassion etc. just he tried to change the view of those young students using his zen and human wisdom.
Later I talked to a friend, well established rinzai career person and he was puzzled by Butsuji kancho as well.. then I told him what I could see on tv, and that he really tries hard to get to people with the strong bodhicitta motivation.. well to be closed in the temple behind the screens and to be shike or even kancho, which is a peak of the monk/priest/teachers career may be counter productive. And ex Butsuji man was in plain kind of samugi, shaved head, no brocades person... no fuss and no anything.. I still cannot forget this appearence. Anyway to make this terribly long story short, some of the shike or even kancho are pretty much characters. But generally the system of temples, education shike etc. is pretty rigid.