elmayerle, I agree, I'd have loved to have seen DACT between the Crusader and the Gripen. Especially the J 35F and F-8E/J circa 1969, for example.
Volkodov, the F11F is a neat (and very pretty) aircraft, but with even less going for it than the flawed F4D. With a new engine and modifications it could have been a winner, but that's outside the scope of what I'm looking at here.
M.A.D, the Mirage IIIO as built would totally qualify. No issues, whatsoever, outside of its fairly late date in that period. It would only be in service four years until its replacement came on scene. And you'd still need to decide what to use between 1956 and 1964. The V-1000 is too late and a not a production aircraft anyway, so way out of scope.
Old Wombat, that was actually my original plan, and I still think it's a totally workable, very practical, and very affordable. That's really a philosophical question, though, and in many ways makes the question harder, not easier. If you decide that—since all combat's going to be subsonic anyway—you don't even need a supersonic fighter, then you have to consider the Sabre (in its various forms), the Hunter, Mystère IV, MiG-17, etc. So, why wouldn't you go that route? Well I was reading about supersonic vs. subsonic fighters over the Middle East, Vietnam, and India/Pakistan. Long story short, if both pilots know what they're doing, the faster fighter controls the engagement. He decides when (and if) to fight and the subsonic fighter is largely at its mercy, just hoping the faster one will make a mistake he can exploit. While that may work in a purely defensive scenario, that doesn't lend itself to air superiority/dominance, regardless of whose territory you're flying over.
[Mirage, Lightning, F-8, Draken, F-5]
Thanks for the response, Rickshaw! Let's unpack it, from simplest to most complex. I
love the F-5, especially the F-5E. It was a fantastic plane and—I think quite underrated. Had it existed in this timeframe, that'd unquestionably be my choice, but the F-5E didn't enter service until 1973 at the earliest. Way too late. Even the F-5A didn't enter service until 1964, which doesn't give it a whole lot of time before the F-4E comes online. Same issue as the Mirage IIIO that M.A.D was mentioning. On top of all that, the F-5A was really a fighter-bomber. More comparable to the Mirage V, G.91, or A-4. Great, efficient little attack or multirole plane, but not really air superiority until the F-5E.

As for the Lightning, it never carried Sidewinders at any time in any configuration, did it? I'm convinced that the Firestreak and Red Top (especially) were better than early Sidewinders in a number of ways, but I do worry about their ability to be employed against maneuvering fighters in close combat. The AIM-9B is the only AAM of 1960 that I'd be willing to bet my life on, and I haven't yet come across anything yet to disabuse me of that notion. No matter what, though, the Lightning is out of consideration because—until the F.6—you had to choose between fuel and guns. The problem is that you need both. The F.6 doesn't enter service until 1965-66, and by then you're almost at the F-4E, so what's the point? It's really not a contender.
The Draken's radar issues were fairly common for the day, as you note, but the earlier model had the same radar as the Mirage III, so you can't rate it any lower. The later Swedish radars were superior to the French one, too. In short, I don't know that radar performance was a particular strong suit of the Draken, but once they were installed, I don't see how you can rate it any lower than the Mirage III in that category.
You say the F-8 was a dog in many ways, care to expound on that? That's not what most of the pilot accounts I've come across say. They definitely liked the power that came with the later -16 and -20 versions of the J57, but most of the complaints seem to deal with the power available in tricky carrier landings and takeoffs in the tropics, a situation that it's hard to compare with the land-based fighters on this list. The lack of a BVRAAM in that day wasn't much of a disadvantage since the ones that were around were pretty much terrible. The ammunition feed was a real issue, and I think the biggest letdown it has compared to the Mirage III and its excellent 30mm DEFA guns with plenty of ammo.
Finally, mind pointing to some of the accounts of the Mirage IIIO vs US F-5 Aggressors? I can imagine the Mirage III may have some advantage at high speeds, potentially, but that'd be about it, I'd think. Israeli pilots seemed to rate the Mirage III as comparable to the MiG-21, with the Mirage III having the advantage in the horizontal, at high speeds, and at low level. They generally rated the MiG-21 as better in the vertical, high altitude, and low speeds. I've read similar things from Indian and Pakistani pilots, too. In fact, Pakistani pilots that flew both the MiG-19 and Mirage said the MiG-19 was far better in the horizontal plane than the Mirage and would try to get the Mirage in a maneuvering fight in DACT, where they knew they had the upper hand.
Furthermore, US aggressor pilots flew both Kfirs and F-5Es and considered the F-5E to be the more maneuverable of the two. In fact, the Kfir was chosen because its handling characteristics were similar to the MiG-23 (
aka not agile).

Finally, I'd recommend reading the interview of the
Ejército del Aire pilot Gonzalo O'Kelly on the Hush-Kit blog.
Hush-Kit: Mirage Pilot Interview Parts 1-5What was the Mirage like in the following ways:
A. Instantaneous and sustained turn rates
“Well, not very good at instantaneous- but better in sustained turns as with everything else, with the nose down.”
B. Agility
“Hmmmm, next question please.”
C. Climb rate
“Good enough in those years.”
He says they wouldn't dogfight in close with the Mirage F1 because they didn't have the agility to match it. In fact, he compares the Mirage III to the F-104 in terms of handling optimization, which matches what I've read in other places. Again, no slouch, but not in the same category as its subsonic predecessors, an F-5E, or any of the Teen series, for instance.
Cheers,
Logan