The whole shtick with unscrewable atomisers to refill bottles was a thing in the 70’s upon the introduction of natural non-pressure sprays as one of their selling points, being more of a value to refill (also there were cartridges too).
Then this went away and sprayers became crimped on because of convenience to just throw away/rebuy the bottle (and probably faster for mechanized manufacturing).
Now it’s back due to customers wanting environmentally- conscious options. In the interim, all those crimped-on bottles that get impossible to use if their mechanism fails, as happens with older bottles, or folks wanting to refill them.
I mean yeah, a syringe could work, just getting past the ball mechanism is a pain in the ass. Most folks opt to cut off and decant if there’s something worth saving. Adding in alcohol to dried oil that’s been evaporated due to a bad seal is a gamble.
Considering the going price for By Man, probably worth the risk at this point.