It is the miners who decide and if the economics of mining get too poor without block rewards that they are contemplating closing the shop they do have incentive to change things so that block rewards continue. It will mean a fork but do you really think that any meaningful percentage of miners will stay on the chain that is uneconomical to mine or do they also join the majority on the more popular chain that also keeps handing them bitcoins? The change to happen in the first place would of course need an agreed consensus of >51% of miners that they do want to change.
Also I think the opinion that the dent in reputation of bitcoin will cause everybody to move to an altcoin is the most silly opinion of them all. Which altcoin? You mean some of the thousand different altcoins? If there for some weird reason could be a selection of only one altcoin that everybody would switch to that also means practically the same thing as increasing the supply of bitcoins. What reason there would be to believe that if this ever happens that there would not be a second change from the first altcoin to some second altcoin. This in my opinion would dent the reputation of all cryptocurrencies irreparably. Changing bitcoin protocol would dent the reputation of bitcoin for sure, but the alternative approaches like changing to some altcoin are much are worse for the overall reputation and so the incentive of everybody in bitcoin economy would be to show happy face to the change and hope the dent do not last for long.
It is even possible that there is no dent at all and with some spin in marketing the general public does not even notice or agree that there was any change at all. The marketing message could be something like: "There was an emerging disaster looming in bitcoin as there would have been a hugely disruptive change deep down in insignificant details in bitcoin halving the block rewards but thanks to heroic effort of tireless developers and majority of bitcoin miners the looming threat was found early and thwarted and bitcoin will continue to work exactly as it has previously worked with no perilous and disastreous changes like block reward halving. Praise the lord, amen."
Actually I have the opinion that there is some nonzero probability of change of bitcoin protocol at some point in future.