The receptionist in Denial is actually the Luck-Based Mission / Only the Worthy May Pass part of the trial.Almost every Holder has a part where some people instantly fail for some reason. Maybe this is why the Holder of Denial's name is... well, the Holder of Denial, since you have to deliberately not follow a certain instruction. The button has to be there for some reason.
The Holders are all technically in the same institution.As stated on the main page, the various stories of the Holders begin with going to any mental institution or halfway house. However, when a Seeker goes into said institution and asks for a Holder, they are taken to a place that seems to involve Alien Geometries. It's quite possible that in the world of the Holders, the institutions are somehow connected through gateways to a massive institution that is housed in an alternate dimension, reconfigurating itself so that the Seeker can find the Holder they are looking for.
The Items are in fact Lost LogiaGoign along with the above WMG, the Holders are some sort of Lost Logia cult that collects these artifacts and then hides them in these alternate dimensions, the trials are purely to find candidates who can activate the Logia, and the killing of those who fail is to prevent anything they may have learned along the way from reaching the knowledge of the Time-Space Administration Beureau.
The stories are Schmuck Bait set up by seemingly clever Seekers as an alternate means of getting the Objects.Any brief perusal of the Holders' trials reveals just how utterly impossible they seem to pass. Anyone who attempts them risks undergoing any number of fates worse than death, and the rewards are minimal (and sometimes are themselves fates worse than death). So why would anyone ever want to go about getting just one of the Objects, let alone all 538 of them? That's what a certain knowledgeable person (or persons) with an interest in acquiring some (or even all) of them asked. So a clever plan was made: why go through the trouble of attaining the Objects, when you can just have some other idiot get them for you? Concisely written instructions on how to acquire each Object have been written and distributed on the internet, with the sole purpose of tempting budding Seekers to set out after them. Then, it's just a matter of finding the Seeker and taking whatever Objects he has, by any means necessary. Mind you, it's not very easy either, but much safer than undergoing the trials.