Not to mention, cold is an infectious disease too (it's literally the same disease, just a weaker variant caused by strains that evolved their potency away), it too will spread to other people if they go to work.
“The cold” is actually any of a wide variety of different viral diseases (caused by various forms of rhinovirus, coronavirus [0], and, I think, a few other kinds of viruses), none of which are flu (influenza virus). It is not a less potent flu.
[0] so calling COVID-19 “a bad cold” is correct from a certain point of view, despite being substantively misleading.
Still, my main point holds - you usually can't tell by symptoms alone, whether it's a common cold or a flu. In case you get severe symptoms, by the time you can, you've already been infectious for some time. So either way, the right time to call in sick is when you first notice the early symptoms - stuffy/runny nose, cough, headache, elevated body temperature. But obviously nobody does that, because it would mean calling in sick at random a dozen or more times per year.
Post-lockdown COVID was the only time I know of in the last few decades, and a brief time indeed, when it was socially acceptable to react to potential symptoms at the right time.
By the time someone is able to tell they have "the flu", they should've been on a sick leave for 2-3 days already.