It depends on the airline. Short answer: European airlines tend to be bad, USA and Asian airlines are quite good. Low cost carriers usually are terrible.
First, no one knows if the flight is really overbooked until the flight is edited at the close of check in a few minutes before departure. Unexpected missed connections, people with visa/documentation issues, people making last minute changes, or even online-checked-in "no shows" can still interfere with the airline's knowledge of whom they expect to be aboard. However at the editing stage the decision whether to confirm or offload/rebook will be made.
Of course in some circumstances it is quite clear that the flight will be overbooked probably by some number of passengers. However, the gate agents are not really supposed to make changes to your ticket, particularly against the ticketing conditions. This is a culture difference. In the USA, the gate agents are usually fully qualified reservation staff who know their stuff, they understand enough of the principles of ticketing, they understand the effect on the airline and they can see how likely it is that other flights are overbooked. In Europe, the gate staff tend to be lesser trained people who should not really be mucking about with a ticket in case they break something.
This is particularly true at outstations where often only local contractors representing the airline are working. In this circumstance, any changes will have to be run through the local station manager or even telephoned back to the head office.
At major stations the real airline ticketing staff can make virtually any change they wish to your ticket, entirely outwith the fare rules, but this is carefully audited. So they will have to be sure they really are saving the company money when they get questioned about it.
European airlines continue to rely on flexibility as a price discriminant, so there is strong pressure to take the risk that things will sort themselves out and deny changes. If you look in the US, many airlines offer "same day standby" to elites in the frequent flier programmes. So it is no big deal for me to change my reservation on the same day, for any reason, when travelling in the US. (Assuming there is still space, which because of the US obsession with free upgrades, there usually isn't, but that's another story.)
It never hurts to ask if there is space available, but usually in Europe, at an outstation with a couple of flights per day, you are wasting your breath.