In particular I'm wondering what SS did with hollows. Did they rely on Quincy since the Asauchi weren't around? Did they just destroy them?
And how did the Soul King perform his duties, without the Shinigami's Asauchi to rely on? Since the shinigami acted as balancers and keep the worlds in check, how was the SK able to act as the "linchpin"?
I think Shinigami could only use Kido and different weapons like Ichibei.
Well, there are several possibilities as the zampakuto itself has many uses and implications. As far as their powers it is plausible they could only manifest them in rudimentary ways if at all. The asauchi helps in not only in bringing out your powers to the real world but also in communicating with it so it is plausible shinigami could not even properly communicate with their zampakuto. Even if they could they would not be able to materialize it. Nimaiya did make the point that no one had ever materialized his zampakuto without an asauchi after all.
Now, as far as the hollow bit it is plausible they simply went with rudimentary methods. If the problem got bad enough they simply killed a bunch of people to tip the balance back. Or perhaps they had a weapon which did not quite work as a zmapakuto but had the function to purify souls. But without the qualities of a zampakuto.
They probably just used old fashioned combat with normal weapons and all. Along with kido ofcourse. Perhaps there were some special ones that could manifest some ability. Ichibei obviously comes to mind.
We also saw with unohana against Zaraki how she used other smaller knives and stuff to get an upper hand which she told Zaraki was how she always fought. There's Zaraki who fought with pure brute force and was still overpowered, so much so that he unconsciously restricted himself. Yama jiis fight against wonderweiss comes to mind, where even in old age, he obliterated the latter using raw fighting skill with hand to hand combat and without a sword.
Coupled with speed and other factors, they could be just as dangerous. Just look at what nimaiya could do using his raw speed and fighting ability and a base sword (although an overpowered one). Yourichi has supposedly been fighting for years without a zan or sword.
There are also forbidden kido (which were obviously used at one time before being deemed so) and those in the nineties level which are particularly devastating.
And ywach refers to the original gotei 13 as a bunch of 'brutal killers' somehow that paints the picture of vicious slashing and what not
. I guess since they had no convenient power at their beck and call, their fighting ability had to be god like and also why they used whatever means necessary to win.