So I feel like people are treating the contest journey comparison as the only important factor between determining who is a better character and also seem to be treating this question like it is a question of fact, rather than opinion. I personally am going to answer it with an opinion (one I assuredly can tell will not be well received). And a quick disclaimer, I absolutely love Dawn, so don't get the wrong idea about that, she is easily my second favorite character in the series history. I just like May a lot better, I feel as though her story is more compelling and I adore her personality. I can admit that Dawn's contest story was better done, but that was inevitable. Contests had a lot of inconsistencies still in the anime during AG because they didn't know exactly how they wanted to adapt them. If May hadn't come first, Dawn would not have had the same compelling story that she did. May was a trailblazer in general as a secondary-main character who actively pursued a goal and if she wasn't so well received, Dawn would have never had her chance to be a co-main character. But just because Dawn got more screen time, that doesn't automatically make her story more compelling or interesting.

What I found so charming about May was that she was quite different than anyone we were exposed to during the first five years of the series, because she was largely indifferent towards/didn't really like Pokemon. This is quite different than disliking Pokemon due to some fear, she just really had other interests. This was our first significant exposure to this in the Pokemon universe, because the people Ash typically encountered were fairly obsessive about Pokemon, but now we see this girl who really doesn't care about them, but just wants an excuse to travel the world. That's one thing people are missing too, she didn't take a long time to find a goal, that was her goal, she just wanted to travel the world. This isn't a goal that was dropped either, she always showed a significant interest in traveling to new places and seeing what those places have to offer. As her journey progressed she was really impressed by Ash and Pikachu's relationship and she began to really feel a bond with her Torchic, and over time she really started to like Pokemon. I think they went this route with her character because Pokemon was beginning to lose popularity, and I think they found a creative way to show people it doesn't have to be all about competitions and that anyone can find their own love for Pokemon in different ways.

One of the other things I really loved about May is that she was awful at pretty much everything at first. Beyond being just generally really entertaining, it made her struggles more believable and compelling. May's struggles were more interesting than OS (or BW for that matter...) Ash because she had every excuse to have no idea what she was doing and her lack of knowledge was much more universal and consistent (and adorable). In comparison, Dawn pretty much knew what she was doing, which is fine, I have no problem with them being different, I just was more enamored with/entertained by May's lack of Pokemon knowledge and natural ability. Beyond the more obvious mistakes, I liked how she made some more subtle mistakes too, like overlying on Beautifly and accidentally not giving Torchic enough attention. This was a rare instance of showing a trainer struggling with dealing with splitting attention between a large group of Pokemon, something that realistically more trainers should struggle with. It was also really interesting to see her unfounded overconfidence in various situations and how that was resolved.

Dawn's contest story benefitted greatly because of May's. DP didn't have any inconsistencies in contests and they were able to hit the ground running because they knew exactly what they do. Dawn's rivals are even kind of rehashes of May's rivals. Zoey is pretty much just a friendlier Drew and Ursula is a less cunning Harley. They even used long term planning to improve the embarrassing childhood story (obviously with May that wasn't planned from the beginning - and it also didn't really impact her/scar her like Dawn's did). Point being, Dawn's contest journey being better was completely inevitable (although I also don't think every change was an improvement, many were). I just don't think Dawn's personal development was quite as interesting, she always knew she wanted contests so she was more focused on contests in her spare time, whereas May was less focused on training all the time leading to being more laid back and fun. As far as personality differences between the two, I just prefer May, I don't pretend to say she was factually more interesting or anything like that, but I would wonder how other people would answer this question if they weren't so overly focused on the contest comparison and just considered the two of them as a whole.

Dawn was arguably a better strategist than Mary Sue was, at least as far as Appeal move ideas and move combos are concerned. Unlike May's obsession with Squirtle's Ice Beam for Appeals in the BF arc, Dawn utilized all of her Pokemon for Appeals throughout DP quite consistently.

I can't even comprehend the Mary Sue insult, out of all the main girls who could be called that....May was pretty much bad at everything she did at first throughout the majority of the series (including contests), while in comparison Dawn was quite good at a lot of things she tried right off the bat, she just had her losing streak in contests, that was pretty much the only thing she struggled with. In AG, the series that also brought us thunder armor, last second annoying wins were just a reality of the time. That shouldn't be a reflection on May alone, because it was just how the series was in general. Plus they didn't hide the fact that May was fairly lucky in a lot of instances, it was made quite clear that spamming assist, metronome, or silver wind was a bad idea. That was an interesting twist, because she actually failed to learn as much as she should have by relying on luck, which in the end cost her in the grand festivals. It is a less straight-forward storyline, that actually was quite interesting. And May realized she wasn't creative enough, which is why she decided to go off on her own, so its not like they were trying to say she had completed her improvement. In contrast, at the end of Sinnoh Dawn decides to stay in Sinnoh for nothing to do with contests (although it was noble of her to do something that her Buneary really enjoyed).

And I find it funny that you mention Squirtle overusing ice beam, because although I don't disagree with you, I seem to remember Dawn over relying on the same move in many instances with Buneary, who also never even learned a new move in a greater than 180 episode run as her second most often used Pokemon.

They would even cut out May’s contests to focus on Ash training Swellow or snorunt evolving to Glalie rather than focusing on her battles.

I think this was more a reflection on Ash and his lack of maturity at the time to where he didn't really support his friend like he should have. He hadn't yet gotten to the point where he could watch a contest and actually learn and apply that to his own battling (which he later was able to do) and he was just too selfish at the time to lose out on time for training. Having him be more supportive of Dawn was an interesting bout of inter-series character development, especially when they showed Paul outside of contest halls a couple of times making fun of them, because it was somewhat reflective of what Ash would do in AG (without the negative attitude, of course).