How do you transit back to training after illness? Do you go straight back to training at full speed or take it easy the first few exercises? I am thinking of a regular flu etc. that last less than a week rather than any major medical condition that would result in serious deconditioning. Do you have a strict policy on not letting people train while sick or handle it case by case?
Gym policy on training when sick is "don't" and while some of this is related to health and speedy recovery mostly I do not want one person's illness to be transmitted to everyone else. I say keep it to yourself. We are all touching the same bars and mats, breathing and sharing the air, quarantine is not bad in this case.
Training frequency is important so, depending on the type of illness, continuing to train while sick may lengthen the recovery time, which ultimately reduces training frequency. Why add to the body's stress by training when sick? Most of us could stand to take some time off anyway so "enjoy" the down time. Make use of it by learning something, if possible.
If you do get sick (flu like you mentioned) then once cured, return to training gradually. There may be some mucus remaining in the lungs that limits top-end performance. There may be some stiffness due to a lack of movement. When you view your training and development over a long enough time-line there is no need to get right back into full speed training (unless there is a time limit established by an important upcoming competition or similar). For most of us step-by-step is best. Your body will tell you what it needs.