It's not that nobody should be able to reach them. As an sysadmin role myself at the mo (I've worked in most jobs in IT over the years), it's a case that I've only got time to field a limited set of things. These are the things that change the big picture in the infrastructure, and that's what takes most of my time.
I'd like to be able to help out more with the individual systems, seriously... The techs that get to go out and fix the small problems are the knights in shining armour; they get to fix the smaller mistakes that users get themselves into (oops, I accidentally deleted some files, oh my PC works again now you've fixed it, so on).
The people that do know me are the heads of departments; they filter in requests that make a business sense to them, and request that they be implemented as a technical solution. Things relevant to the business in the wider scope make it to me.
When I took on the role, it had an inordinate amount of calls from users who wanted to short circuit the help desk (no logging means we can't prove we've done the work to the accountants for a start). Everyone's work, to them, is top priority, after all, it's they who are affected. It took a while to get that under control, and even to people who I consider friends in the organisation, if it's one PC that's affected, it really isn't my problem. If a thousand are, it probably is my problem.
To run a company, roles need a frame of reference. Some make the mistake of believing their frame is the whole of everything that is (hint, it's not). The further you work from your core frame, the less effective you are at doing the core work. If you find your strengths are in a different frame, you're in the wrong job, so change that.
Assuming you should be able to go direct to the admin assumes you know the technical impact of the problem you have (in the enterprise wide scope), know exactly how to describe it, how it's impacting every other system, the amount of users affected and a whole host of things (which is a picture that's built up by the Helpdesk and escalates through the technicians). If you've spent time doing that, what have you been doing in your real job? There may be many people with your level of skill also phoning the help desk, and they may have different views and conclusions based on a different geographic/business perspective.
Doing things the right way lets an accurate picture be built. If all 5k+ staff phoned me in a huge incident, I'd neither be able to get a real picture of it, communicate with the people I needed to, nor actually talk to most people. I'd also not be fixing the problem, which is the real kicker.
Incidentally, HR does work that way; it's the only way they can research the query, and give me an accurate answer that lets me work on a factual basis (rather than "Oh, I seem to remember that it's something to do with X. Probably. Bye then."