The process for getting a clearance is long and it's expensive. If the company has work that needs to be done in three months that requires someone with a clearance, there simply isn't enough time to get someone cleared.
I don't mean to bicker over semantics, but this statement of yours does equal narrowing the pool of applicants, as I said. It narrows it to people who could do the job in the time required.
When I applied for jobs with various agencies, including the CIA, I was specifically told that requiring clearances already was used as a weed-out when they expect an onslaught of applicants. When the jobs required were so specialised they didn't think they'd have many applicants, they could leave that requirement off.
Similarly, departments at various universities stick 'Masters degree required' to narrow the pool of applicants, even when they have desk jobs that could be done by people who don't need Masters (or in many cases, even Bachelors) degrees. At Indiana University, I was once asked by the head of a department to apply for a job. I hadn't completed my Masters degree at that time. I tried to apply through personnel, and they wouldn't take my application because the application prerequisites showed 'Masters degree required'. I asked my friend why he put that -- his response was (and this is a quote), 'Well, we had to put something.' The personnel office had no way of knowing that a Masters degree wasn't really required in that department. They just went by what they were given on the forms.
In the end, the department sent down a revised set of requirements. I still decided not to apply.
I'd say apply for anything you can, and see what happens.