People who remove flowers to collect pollen from the next day, do so because they have other plans for the evening, or because they don't have permission to work overtime, or because they can't stand mosquitoes, or because they are afraid of the dark, or ....
I gather pollen when it is shed naturally and use it immediately. If it is dark out, I wear a hiker's headlamp.
Female flower parts that are "pregnant" because they have been pollinated are not able to suppress the male parts from going ahead and being male, as they always do. The female parts are normally no longer receptive by the time the male parts in the same flower are ready to do their thing, so there would be no point to impeding the shedding of pollen.
The only time one needs to avoid using pollen from an already pollinated flower is when one is doing planned hybridization and therefore needs to be sure of knowing who the daddy is. So one avoids having any of the previous night's pollen mixed in with the fresh pollen one is using to breed with.
If one is only interesting in achieving fruit set, without a care about the seeds, then certainly, go ahead and use pollen from previously pollinated flowers!