The basic potential problem with filling staff positions in this way is that there is no guarantee that you are going to hire the most qualified person for the job. This would be less of a problem for a job that requires very little in the way of skills, but it could be a problem even then.
Imagine if you are hiring carpenters. You hire the first person who walks through the door or you hire whoever has their name come up in the lottery. There is no guarantee that you are getting someone who has ever held a nail gun, let alone the absolute best person for the job. Jobs like this, which require skill, cannot be filled on a random basis.
This sort of a staffing process would be less of a problem in a less-skilled job, such as a job as a cook at McDonald’s. Anyone can be taught to do this job (I have done it myself, so this is not a slam at people who work there). However, it is still not a good idea to simply hire the first person who applies without at least interviewing them. What if the person has been fired from four jobs for being lazy? What if the first person comes to the interview drunk? These are extreme cases, of course, but they are things that could happen if you hire people on a random basis.
Both the notions of hiring on a lottery basis or a first come, first served basis assume that workers are interchangeable. Human beings, though, unlike machine parts, are not identical. The first person who applies for a job or a person selected by a random lottery does not necessarily have any relevant qualifications and may even be a potentially bad hire.
Although for some forms of temporary unskilled labor such as shelf stocking after hours in a supermarket, a lottery process might work, for most jobs, one wants to choose the best qualified applicant. Even among applicants with strong qualifications, there are differences in fit. For example, a start-up tech company might be looking for a different type of person to help design a new product than a larger company hiring someone to upgrade an existing product and a company working primarily in government and defense contracting may have different priorities than one creating consumer products, even if both companies are using similar technology.