To borrow the words of the Head of Square-Enix, "Points for originality, but I doubt it'll actually work."
You ask for more reasons on why people are so 'adamant' on keeping the damage meters on in a tournament-style match, so I'll do my best to be as thorough as possible. Note that I'm probably forgetting some things.
1. Prediction of Strategy
A large portion of Smash Bros. strategy is based on prediction, sometimes commonly referred to as 'mindgames'. The ability to predict your opponent's actions comes from a variety of places and for many reasons (trapping the opponent into a cycle, noticing canned or repeated responses to situations, knowledge of move properties and degrees of knockback, etc.), but a large indicator of certain opponent's actions is the damage gauge. When at low percentages, it is prudent to take bigger risks, as punishment at low percentages is difficult. As a player's damage rises, though, many players begin to act with more reservation instinctively and, as such, take fewer risks (or take actions will lesser risks more often). This is a primary basis for the concept of 'pressure', as well (among other things such as spacing). Being able to discern, at any moment, the exact numeric advantage one player has over another is a very large part of being able to predict any player's actions and is a staple of gameplay. As a correlation...
2. Creation of Strategy
Being able to discern degrees of difference between player damages allows players to craft strategies on the fly that take advantage of specific situational details, rather than promoting strategies based off of properties common to all situations. This important detail creates dynamic fights that force players to act more within the constraints of the battle at hand. Many strategies are dependent on difference in damage (such as the above mentioned 'pressure' strategy), and cannot be reliably or consistently applied without the important knowledge that damage meters provide (after all, its difficult to pressure someone into an action if he/she doesn't know he's being pressured, as the very concept of 'applying pressure' relies on a player reacting to a situation with reservation; if a player doesn't know that his opponent has a commanding lead, then he will act with the same bravado that is used at early stages of a fight, in theory).
3. Reservation of Moves
As has been mentioned before, moves degrade in power the more often they are used, which can be problematic for a game with no discernible way to distinguish amount of damage (that it to say, no way other than after the fact). Being unable to preemptively know when a kill move will be successful can have one of two outcomes, one more likely than the other. The first (and less likely outcome) is that players will make sure to reserve kill moves and strategies for only when they are sure to work (which would be a positive, I'm sure), but the second (and more likely) scenario is that people will end up using kill moves too often, as waiting until 100% certain to use a strategy blindly (or next to blindly) is disadvantageous (the more time an opponent is on the field, the more time he has to counter-attack you). Even if this wasn't the case, damage meters also enhance the preservation of kill moves by showing both players exactly when they are sure to work, which can easily be linked back to our first point.
4. Combo Viability (a la pockyD)
As pockyD mentioned earlier, damage meters have the inherent advantage of promoting strategy (and prediction) by indicating exactly when certain combos are able to be used. For a real-world example, I play Toon Link and use a (very basic) three-hit combo consisting of a dair linked to a second dair (by bouncing) linked to a fair or bair (depending on spacing). I find that this 'combo', however, only works between 30-65% or so (depending on how high I can bounce) because of the amount of knockback the second dair strike gives on the ground (TL's dair acts as a meteor); as such, I cannot simply use the 'combo' all willy-nilly, and must reserve it for when it is strategically useful, something which is hard to do without accurate knowledge of damage amounts.
I hope these clear up some of the reasons why we play with damage meters on; I'm sure that there are inherent benefits to playing with them off, too, but as far as most people (myself included) are concerned, the opinion is that the pros are outweighed by the cons.
