and Con is hardly any less useful.
I disagree. Under a maximum hit point rule, Con loses a lot of punch, especially for classes with higher HDs.
A 4th level fighter with a Con of 14 and random HP rolls will have approximately 35 hit points (10 + (3x5.5) + 8). A fighter with a Con of 10 would have approximately 27 hit points.
A 4th level fighter with a Con of 14 and maximum HP rolls will have 44 hit points. A fighter with a Con of 10 would have 36 hit points (which is still 1 more than the random HP rolled fighter!).
A 4th level wizard with a Con of 14 and random HP rolls will have approximately 20 hit points (4 + (3x2.5) + 8). A wizard with a Con of 10 would have approximately 12 hit points.
A 4th level wizard with a Con of 14 and maximum HP rolls will have 24 hit points. A wizard with a Con of 10 would have 16 hit points (4 less than the randomly rolled HP wizard!).
Clearly Con remains important for low HD characters, yet decreases in value for large HD characters. This is actually the REVERSE of what you think of when you imagine the classes. The typical wizard should have less Con than the average fighter. Yet, using a maximum hit point rule, I think you will find that the lower HD classes will be more inclined to raise their Con than the higher HD classes.
Alternately it is clear that this rule favors higher HD classes. In your group, a wizard would be indifferent to max hp every level, whereas a fighter would be thrilled.
Another system would be as a reward for good role-playing or otherwise enhancing the D&D experience (if your group is focused on combat, for providing tactical advice). You could offer someone an additional roll on their next level's hp, or for particularly ingenious playing, a full HD for one level.