It seems to me that EPS is just earnings divided by an arbitrary number. Why not look at operating profit as a percentage instead?
(Earning / Price) is not a typical metric, (Price / Earnings) (i.e. PE Ratio) is the more typical metric. These metrics are meant for casual/institutional investors not lay people[1].
Meanwhile, if you still want (Earning / Price), you can do (1 / PE Ratio).
[1] If you aren't able to find the number of STOCK units you own or will purchase, then you are not a casual/institutional investor.
Of course that's only a superficial analysis. You have to take into account the book value (asset value). If the company is high growth. What industry it's in. Whether a change in earning per share is due to a one-time event. Etc.
Yes, if you're curious how well a particular company is doing over time, then it's arbitrary and not very helpful. But if you are comparing the relative value of a single share of company A vs company B, then it's helpful.
(Which is then also related to the sibling comment about dividends).
The number of shares you own is not arbitrary - it's actually a very important number to you. So you multiply how many shares you own, with EPS, and you know your personal profit.
In particular your profit per how much that share cost.
Obviously investing in stocks has other ways that you can make money, but EPS is pretty core to the whole thing.
Probably the EPS metric made more sense before Excel.
If you want an overalls figure look at their financial statements.