D&D has an explicit goal (i.e. looting treasure from dungeons), and one's accomplishments towards this end are clearly reflected in their XP total, which could easily work as a sort of score table. How about emphasising this aspect of the game?
The main goal of the game would be levelling up without dying. Achieving any particular level would earn the players points equal to said level squared (i.e. 4 points for lvl-2, 9 for lvl-3, 16 for lvl-4, etc.); losing a character would deduce a number of points equal to the inclusive sum of 1 to the lost character's level (thus 1 for lvl-1, 3 for lvl-2, 6 for lvl-3, etc.).
I would further add that the death of a henchmen also result in the loss of 1 point, thus hiring henchmen became a more complex tactical decision (i.e. not only monetary cost and number of henchmen were to be considered, but also the possibility of losing a few points) and they became more than simple meatshields. To compensate for this, I would probably give each henchmen a particular skill or asset that make them useful companions.
In order to make this work and the pace of earning points matter, the Referee needed to introduce some sort of constraint on the length of the campaign, possibly something that is independent of the players. In many board games the number of turns is fixed at the beginning of the game; Beedo's Black City Project comes to mind as a good example of implementing such restrictions that make sense in the fiction: the eponymous megadungeon is so far to the north that gold-diggers need to leave the site before winter comes and prevents their journey to home.