At low volume the problem it solves does not exist. Twenty applications for one role is manageable in a spreadsheet and a folder. What an applicant tracking system actually addresses is many roles, many applicants, several people reviewing, and the coordination failures that follow. Buying it earlier means configuring a system to manage a volume you do not have.
There is one argument for structure that applies even at one hire, and it is not efficiency. Keeping a consistent record of who applied, what was asked, and why each decision was made is what protects you if a rejected candidate ever raises a question. That record can be a spreadsheet with a row per applicant and a column for the reason, provided it is written at the time and phrased in terms of the requirements rather than impressions.
What matters more than tooling at this stage is the job description and the questions. A specific description filters out most of the mismatch before it arrives, which reduces volume more effectively than any software processes it. And asking every candidate the same small set of questions makes comparison possible and keeps the process defensible.
Revisit when you are hiring continuously rather than occasionally, when more than one person is involved in deciding, or when you are losing track of who you replied to. Until then the free posting tools plus a structured spreadsheet is the correct answer rather than a compromise.
Decide how you will respond to people you do not hire before the applications arrive, because that decision gets made badly under time pressure. A short reply to everybody who applied costs minutes and is the difference between a candidate who speaks well of the process and one who does not. In a small market like this one, where people know each other, how you treat unsuccessful applicants circulates.
Write the job description as a filter rather than a description, since it does most of the work that software would otherwise do. Being specific about the actual tasks, the level of experience genuinely required, and the salary range removes most of the mismatched applications before they arrive. Vague postings generate volume and the volume is the problem people buy software to manage.