If a user accepts an answer, then the question instantly becomes less popular.

It will instantly attract less new answers. But that acceptance is still a valuable indicator to future readers of the question, and the accepted answer in most cases will gather more up votes than the other answers.

Just because the acceptance rate isn't published doesn't mean it isn't there and cannot be used by system as part of its automated functions.

The user mentioned as an example seems to be an oddball, there is nothing you can do about that, they will arrive at the gates in all forms no matter how much you try and stop them. Changing a well established function of the site just to accommodate these people is a total waste of time - possibly the only action that should be taken for serial non-accepters is to assign ownership of their questions to Community so that they don't get any reputation gain from them (not that many of them accumulate much rep anyway).

Should we have an incentive to accept?

The negative incentive didn't really work (where the accept rate was published and others would pester the OP via the comments). But what makes you think that a positive incentive would work, especially as most positive incentives are a target for gaming? Personally I think serial non-accepters should be tracked either via a review queue where their questions get a little extra attention and massaging from the community, or as one of the factors used to identify users for temporary question bans.

If a user accepts an answer, then the question instantly becomes less popular.

It will instantly attract less new answers. But that acceptance is still a valuable indicator to future readers of the question, and the accepted answer in most cases will gather more up votes than the other answers.

Just because the acceptance rate isn't published doesn't mean it isn't there and cannot be used by system as part of its automated functions.

The user mentioned as an example seems to be an oddball, there is nothing you can do about that, they will arrive at the gates in all forms no matter how much you try and stop them. Changing a well established function of the site just to accommodate these people is a total waste of time - possibly the only action that should be taken for serial non-accepters is to assign ownership of their questions to Community so that they don't get any reputation gain from them (not that many of them accumulate much rep anyway).

Should we have an incentive to accept?

The negative incentive didn't really work (where the accept rate was published and others would pester the OP via the comments). But what makes you think that a positive incentive would work, especially as most positive incentives are a target for gaming? Personally I think serial non-accepters should be tracked either via a review queue where their questions get a little extra "attention" from the community, or be one of the factors used to identify users for temporary question bans.