Bale exhibits dominance over Walcott in every category.So, according to a bottom-up view, Bale is clearly the better player. That said, as always bottom-up numbers are difficult to interpret. No complete model of football exists and therefore it is not obvious for each statistic if actually a high or a low number is good. Sometimes the answer is "it depends". In Benjamin's analysis, for example, a high ratio of a player's goals over the team's total goals is seen as positive as one interpretation is that he is a main contributor to the team's success. However, an alternative explanation is that the player cannibalizes goals that team mates could score, too. He maybe only seeks many shots and thus takes opportunities from the team mates. A third interpretation is that the passes he receives are so good that any player could score. So while he does a good job in scoring, he could be easily replaced with a team mate without the number of scored goals changing dramatically.
I have no idea which of the three explanations is the correct one. It could even be a mixture of them or some more other reasoning can be found. This is a general problem of the bottom-up approach. It delivers valuable insight and we can learn a lot, but it is hard to base a player comparison on it because we are sure to miss something.
So let's see what Goalimpact says to those two players. Goalimpact simply measures the player’s team’s goal difference when the player is on the field. It doesn’t look at how he effects his team’s goal difference. The value is corrected for other influences on the team's goal difference to increase accuracy
- the players alongside him
- the strength of the opponent
- whether he has home advantage
- red cards shown to either side
Top-down sees Walcott as better than Bale. The disadvantage of bottom-up is that we can't really tell which player is superior and the disadvantage of top-down is, we have no clue why Walcott is better than Bale. Maybe he is working more for defense. Maybe Bale is cannibalizing goals from his team mates. Maybe something completely different.