I can't say much about Rachel since I haven't played any NG games (since NES days), but I feel that Hayate is being undersold here a bit. The general ninja order for the 4 main ninjas to me are: Ryu > Kasumi > Hayate > Ayane, going by DOAD and DOA5 stories. Thoughts on why below.
(I'm also aware the following text shows I put way too much thought into this)
Ryu is the ultimate ninja. Obviously the top here.
Kasumi:
- Yes, she defeated Raidou, but that seemed to be more guts and heroic determination than a skill gap. Raidou easily defeated Ayane twice, even one-shotting her the second time. But Ayane still forces Kasumi to fight seriously, even winning in the third tournament. The still amnesiac Hayate/Ein managed to catch Kasumi off-guard and KO her - Kasumi obviously wasn't going all out, but getting KO'd like that still suggests a small skill gap, not a large one. The fact that Raidou won in short cutscenes may be more for story pacing and buildup. Still, Kasumi's win over Raidou shows off her raw potential.
- Kasumi's raw speed is amazing even for a ninja. In the Hayate and Ayane vs. Kasumi vid, we see Hayate and Ayane go into superspeed mode at the start. We then see the two in slow motion, and Kasumi's superspeed is so fast it looks like superspeed while the others are seen in slo-mo.
Still, the moment Hayate crossed blades with her, she's put on the defensive and apparently couldn't find the footing to go superspeed again. Her blade was shaking more in that deadlock, showing that a Hayate vs Kasumi fight is a power vs speed battle. (That was the fake Kasumi, but she did beat Alpha-152 earlier, and I'm assuming the clone is equal to the real Kasumi)
Hayate:
- On the oil rig, he fought the false Kasumi unarmed and was implied to be winning before the choppers interrupted. This Kasumi beat Alpha-152, so it's hardly a weak knock-off. In the 2v1 sword fight, he didn't get hit once (unlike Ayane) and was the one to force Kasumi on the defensive as well as getting in the first and final cuts on her. If Ayane wasn't there, it wouldn't be so easy obviously, but Hayate seems to like using his sword any chance he gets in cutscenes, unlike Kasumi. I don't think it'd be an easy win for Kasumi in a real duel.
- Aside from Torn Sky Blast, Hayate can also quickcast (in midair no less) a ninpo that can one-shot a helicopter gunship which he also used against false Kasumi.
- Any time he loses his cool, he does poorly, seen against Raidou, against Ryu in DOA5 (but not DOAD) and even Rig. It also indirectly led to his two captures; the first time he was in a coma from the ninpo duel. The second time had less to do with the fight w/ Rig, and more because he was in a lab that could reactivate Epsilon.
Ayane:
- I think she's the weakest, at least in close up combat, but as a branch member of Mugen Tenshin I'm not sure that's supposed to be her forte anyways.
- None of the cutscenes show her using fast ninpo like Ryu or Hayate. On the other hand, she did use the large scale ninpo that took out a good portion of one of the DOATEC towers. I'm pretty sure her role is support (demolitions/tracking/etc) rather than up-front fighting.
- She lost against Kasumi every time except once. In the 2-on-1 fight against false Kasumi, she didn't land a single blow until Hayate did, and Ayane was hit with a full combo before that. Even in DOAD, she was surprised that Hitomi held her own with her karate. Ayane was also the most exhausted in the fight against the final Alpha clone.
Again, I honestly don't know if Rachel would win or not since I haven't played NG, but Hayate's hardly a pushover even by ninja standards.