Why would one use EPS and not P/E?

I am probably missing something, and will appreciate your help with my beginner question.

My understanding is that P/E represents the ratio between company’s total value and company’s total earnings. This sounds like a very simple and reasonable metric that shows the relative profitability of a company and it can be used to compare different companies.

EPS also represents the ratio between earnings and total value, but the ratio is also divided by the share price, which by itself doesn’t hold any information about the company’s performance. So EPS can’t be used to compare different companies.

I get it that I can use EPS to track company’s success over time, but I could also use P/E for that and get an extra benefit of being able to compare different companies.

Does it make sense, or am I totally wrong somewhere in my understanding? And why then is EPS so popular, for example in companies’ earning reports?