How Do We Judge These Days?

We saw last time that in order to force someone to adjudicate, it is necessary to have dayanim who have authentic semicha that was passed on from the time of Moshe Rabbeinu. Considering that the chain of transmission has been broken, how does the beit din system work nowadays?
The gemara (Bava Kama 84a) says that nowadays we judge people as agents of the last dayanim semuchim. The gemara in Sanhedrin (3a) gives a somewhat different explanation, claiming that it is a rabbinic institution so that "doors will not be locked before those who need to borrow money" (in other words, a necessary step to ensure that a normal society can continue). Why is this necessary if we have the idea of serving as agents of the semuchim?
Tosafot (Sanhedrin 3a) says that both ideas refer to the same rabbinic institution. Rashi (ad loc. 13b) says that one institution discussed a time when there were semuchim but not in the specific place, and the other referred to a time when there were no semuchim anywhere.
In any case, the gemara limits the use of non¬-semuchim judges to cases that are common and involve a loss of money. Two examples of cases that are excluded are damages to a person’s body and payment for embarrassment. The Rambam (Sanhedrin 5:10, accepted by the Shulchan Aruch) says that regarding damages to a person’s body, while we do not judge depreciation in overall value, we do judge loss of income and medical bills. The S’ma explains that smaller damages that do not affect one’s overall value but do cause absence from work and medical expenses are common. The Rosh (accepted by the Rama) makes no such distinction.
The Rishonim discuss how it could be that the rabbinic concept of agency for the semuchim of old can work to force a get and to do a conversion, which are Torah laws. Some say that in fact the ability to continue these practices is rabbinic, and they say that the absence of these processes would cause great damage, thus justifying their implementation.
Rishonim point out that the minhag of the ancient communities of Bavel was to excommunicate those who did not appease their counterparts whose cases did not warrant a beit din hearing due to lack of semuchim. Furthermore, when the two sides agree to adjudicate upon that which beit din does not have jurisdiction, their agreement empowers the beit din. We would also point out that many cases of bodily damages are governed by agreements with insurance companies. In those cases, the issue is not of a Torah-levied payment for damages but is based on prior agreement and this is under the jurisdiction of even a contemporary beit din.