The answer is, as always, that this depends on style of play, and also on the edition of D&D.
Easy with modern D&D
With D&D 4 and D&D 5 there are no mechanical problems with the suggestion.
Takes some thought with older editions
With D&D 3, characters can use experience for creating magic items. Pathfinder is an example of how to change this, should one not want to use experience points. Unfortunate multiclassing also causes experience penalties, but forgetting this entirely is not an uncommon house rule.
With older editions of D&D, the experience that characters require to level up depends on their class, and the variance is large. One would need nontrivial house rules to retain the effects of this on gameplay, while at the same time not using essentially experience points.
Reasonable if following (a scripted) story or playing through disjointed adventures
In organized Pathfinder play, characters gain a level for every three adventures they play through (roughly speaking). In a game where one is following a rigid adventure, or a story prepared by the game master, the GM can simply say when everyone levels up, and there will be very little effect on gameplay.
If following a story or module, levelling up at game master whim can help in keeping the power fantasy going, or in creating actually challenging fights, whichever the game master prefers.
Not so good with OSR sandbox play
In OSR play characters should be rewarded for good play by their players. The typical way of achieving this is by mostly giving experience for recovered coins. The play often happens in sandbox, and the referee may run the game with an open game table, or West marches -style. Characters often die. One player may have several characters.
In this type of game, earning experience and levels is a significant achievement, can happen incrementally, and leaving it to referee whim is against the purpose of play - to measure player skill in a dangerous fantasy environment, with a neutral referee. With often shifting cast of characters, it would be difficult to keep track of who has adventured enough to earn a level. Experience points are a good measure for this.