You play board games to have fun. If the game rules or objectives lead to situations where trying to achieve those objectives isn’t fun, then you need to either change the rules or the objective. In multiplayer games, a player will often find themselves in a situation where they can’t get first place; in the absence of a secondary objective, they can’t have fun, so clearly you need to add something there. I find going for second place (etc) is a good secondary objective that allows that player to still have fun without impacting (too much?) the fun the other players are having. Other games than Dominion will allow “maximizing your score” as a good secondary objective (eg Agricola). Some games, by their structure, don’t allow for secondary objectives, and then I would expect players that can’t win to try to end the game faster, which is usually not fun for anyone involved; but I would say that’s a problem of the game, not of the group.
Of course, this leaves some subjectivity in the hands of the player, since they have to decide when to stop trying to go for first place and focus on second place, etc, which is not present if the objective is “first or bust”, but I think the pros outweigh the cons.