Getting treated with antibodies don't mean you're asymptomatic or that you'll feel well. It only means you'll have less severe symptoms. You'd have healthcare workers walking around sick, spreading the very disease they're trying to treat others for. Or taking up a hospital or clinic bed, not treating patients, using up limited healthcare resources — which, for the duration of their illness, is a worse outcome for their patients than if the healthcare workers weren't there at all.
It's expensive. There's no economy of scale for an unusual disease outbreak. That applies to both antibody and vaccine stockpiles.
If the disease is minor (like COVID-19 usually is for otherwise healthy individuals), healthcare workers might be willing to take a chance to avoid the hassle of PPE, but would their employer (such as a hospital, or MSF)? They're the ones who have to pay for antibody treatment if their healthcare employees get sick. They're the ones who have medical ethicists looking out for both the well-being of the frontline healthcare workers and patients.
If the disease has a significant mortality rate, availability of a specific antibody treatment, even if it's stocked and instantly available, wouldn't motivate anyone to go without PPE. Nor would getting vaccinated. They're not guarantees of survival, and preventing healthcare workers from getting sick is more important than treating patients.