That is to say most people don't have allergies to confuse with illness and although you might wake up feeling poorly for a number of reasons flu sufferers are liable to suffer from one or more distinguishable symptoms for example for myself when I've contracted the flu I always get a really shitty body ache.
It's possible to get a cold without also having a fever. You are absolutely correct about that but even in that instance the majority are liable to be able to know when they are ill. They just don't stay home because the social pressure has been in favor of bringing your sickness to work as if you were doing everyone a favor by showing up and making your cohorts sick.
Sometimes my throat is sore but its just allergy drainage. It doesn't help that the things I'm allergic to line up with flu season.
https://www.aaaai.org/about-aaaai/newsroom/allergy-statistic...
https://acaai.org/news/facts-statistics/allergies
1 Hay fever, or allergic rhinitis, is a common condition with symptoms similar to those of a cold. There may be sneezing, congestion, runny nose, and sinus pressure. It is caused by an allergic response to airborne substances, such as pollen.
2 The American Academy of Allergy, Asthma & Immunology (AAAAI) is the most prominent membership organization of more than 7,000 allergists / immunologists (in the United States, Canada and 72 other countries) and patients' trusted resource for allergies, asthma and immune deficiency disorders. This membership includes allergist / immunologists, other medical specialists, allied health and related healthcare professionals—all with a special interest in the research and treatment of allergic and immunologic diseases.
WebMD's doesn't actually seem to actually source their data at all despite styling it like a link for some reason and since their is a disparity between what the CDC and AAAAI say I strongly suspect they are full of shit