For me, it's two things.
One, many of the folks who talk about restricting (or, quite commonly, "banning"--some even do so rather gleefully) explicitly say they've done so for ages and ages, like 20+ years, and then append that with something like "and I've never seen any reason to change" or "and I've had lots of fun with that" or whatever. So it seems only appropriate to say, "hey, I've had a lot of fun doing things the opposite of that, maybe give it a try?"
Two, I very rarely get the impression from proudly "traditionalist" gamers that they ever really have considered other ways. They often (not always, but often) speak in incredibly dismissive, even uncharitable phrasing (like "Star Wars cantina on feet" or other ways of expressing how yucky and weird physiological diversity must be). They often (not always, but often) react with rather a lot of hostility to alternative approaches, even when suggested in very mild ways, such as a thread I made a while back advocating that random, irrevocable permadeath is a really pretty dull motivator and that others are much more effective. I then had easily a dozen people accuse me of wanting to strip all value and challenge out of the game e.g. "Oh so the players just win all the time forever? How boring!"
Perhaps I am uncharitably generalizing. It wouldn't be the first time and it won't be the last. But I have in general found that there's a scornful attitude toward things with insufficient pedigree (which of course makes a catch-22: You can't get the pedigree without use and representation, and you won't get any representation until you meet the pedigree requirement) or that are "too weird" or whatever. Hell, at least one designer of D&D itself make jokes like this, back during the Next playtest. He openly joked about how he couldn't understand how anyone would want to play dragonborn, with how weird and non-traditional they are, but that he'd very slowly warmed up from "nope, you can't play one in my games" to "well okay, if you really like this weird thing I guess you can play it," and very clearly presented this as a magnanimous and significant shift.
So...yeah. I do get some amount of impression that a lot of "traditionalist" DMs haven't tried, and maybe even haven't considered, these other approaches.
I would take a slightly (but only slightly) more nuanced approach. That is, I'd consult with the group and try to reach a consensus. Perhaps the rest of the group was really excited to play evil characters, so adding that restriction would sap the group enthusiasm overall. Or perhaps the campaign premise kind of requires characters with legitimately unsavory pasts. Whatever the reason, it's quite possible to want to talk it out.
So we do so. Is it "I'm an abuse survivor so a lot of stereotypical 'evil' deeds dredge up painful memories I'd rather not think about while gaming"? The folks wanting to play evil might find that an acceptable reason to choose something else, or that answer might push me to look for a new campaign premise, etc. But it might also suggest that we talk out the boundaries and try to find something that is acceptable to all parties. Maybe "evil" is a cosmic label so you can be morally upstanding but because you're a necromancer (e.g. one inspired by the Abhorsen books) or a half-devil (e.g. Hellboy) you're "evil" by default. Or maybe everyone used to be hardcore evil, but they've begun to reform, and this is a tale about whether that journey is possible for the various characters. Maybe the campaign premise can be reworked for people wrongfully imprisoned. No one side has a unilateral "things happen my way" power, and all participants are free to decide that this can't actually be resolved and just needs to be abandoned (though I would hope they would all bring a good-faith effort to determine that rather than keeping a finger firmly on the big red button just waiting for a reason.)
I had not really gotten the impression, prior to this, that there was much listening to your players involved. That is, proposals made even in tangential hypothetical were met with (paraphrasing) "well I don't do that," without much room for discussion (and attempts to do so tended to then produce "well, I'm the DM, it's my job to make a final call and I've done so.") Perhaps you would face it much differently if one of your actual players had responded thus.
I, too, look for what the player is trying to accomplish, though I tend to be persuaded by "rule of cool" so long as the request does not depend on inducing other players to respond in specific ways (that is, the request must not be coercive), nor on acquiring an unfair or illegitimate advantage not otherwise available (that is, the request must not be exploitative). So if the player can sell me on it being neat or interesting or fun, I'm game, we'll figure something out. Like you, I don't run games for evil PCs, because I heavily doubt my ability to run an enjoyable game for them. Beyond that, though, I try very hard to avoid bright lines, because I am keenly aware of the limits of my ability to predict what the group will enjoy.
For the half-dragon half-vampire, honestly I find the billowy cape thing hilarious so i'd probably allow that element just because I can (and would have people comment on it regularly, e.g. "how is his cape billowing in an enclosed cellar...?") I follow Gygax's requirements for playing more extreme monstrous races: go for it, but you must grow into your power like anyone else does. Given the limit on evil characters, how does this half-vampire feed? How did they become a half-vampire? Are they weakened by something (perhaps having gone a long time without feeding on blood) and need to recover their strength? Is their body actually dead dead, or does the "half" part imply they're fully or partially alive? I'm cool with a mix of both, but that should (at least initially) come with some drawbacks or be more "worst of both worlds" than "best of both worlds" at first. Also, how does their draconic heritage affect them? Can they still use elemental breath or has it changed? Are they more humanoid with just a light touch if draconic elements (e.g. the 3.5e Spellscale race), heavily draconic (e.g. 4e Dragonborn), or somewhere in the middle? Moving from specifics to abstractions, why this specific combination? Is there a particular story they want to see play out, do they just think Castlevania and Skyrim are cool, or is it about mechanical advantages? Does it specifically need to be "the literal child of a dragon and a vampire" or could other origins or possibilities apply? Is the player okay with starting out pretty weak (as far as vampires and/or dragons are concerned) but having potential for some cool powers down the line? Are they willing to cede some control over backstory and relationships/history/past deeds to me, to enrich the experience of playing through the character's story? Are they okay with other players having similar open-ended potential even if those players choose more typical/traditional options? Etc.
As for the Flash tornado, well, what does that do? Is it powerful, or just cosmetic? On its face, the way you've phrased it, it sounds like a naked power grab, but I've no way of knowing how the original request was presented, nor how the person requesting it would have responded to questions. If the player is okay with this starting out as a weak or even purely cosmetic thing and then growing over time into something powerful, I have no innate objections. Sink enough resources into it and it could even become a truly fearsome ability. But if they just want to be able to fly and (effectively) give themselves time stop whenever they like then no, that will not be happening, because that's very clearly exploitative. Others have to work for that power, so this player should have to as well.
