Even when I was looking for my second job out of college, I submitted dozens of applications and there were several times when I was unsure if I had applied for a job or not (not every job portal sends confirmation emails). Once you get into the dozens of applications, you need to take notes to keep yourself organized about if you need to do something (e.g. take a coding assessment), if you're just awaiting something (e.g. awaiting a scheduled interview), or to just take notes about how you felt about some part of the process (e.g. you saw some red flags or positive signs during some interaction).

If you're only applying for a very small handful of jobs (1 or 2), you probably don't need to use a CRM nor something resembling a CRM.