As with most things in photography, the answer is going to be "it depends". Sorry, folks, but there are only so many aspects of a craft that is largely artistic that can be neatly encapsulated into a mathematical formula and presented as "fact".

Big aside here

Even the term "macro" has to be defined rather loosely now. Yes, traditionally (and by "traditionally", I mean in the large-format world), the term has meant "around actual size on the film/sensor". In practical terms, that has meant somewhere between a 1:4 and a 4:1 reproduction ratio. Less than 1:4 and it's "just" photography (in the sense that one doesn't really have to compensate for bellows draw, or light loss due to lens extension, and a little bit of field curvature isn't going to kill your picture). Any greater than 4:1 and it becomes microphotography, which presents an additional set of challenges. Besides the difficulty in achieving and maintaining focus, the greatest challenge in microphotography is that the camera literally gets in its own way. Lenses for microphotography tend to be built in a way that at least theoretically allows the possibility of sneaking some light onto the subject from the front(ish). See this Minolta 12.5mm/2 bellows micro lens for example. Micro lenses for large format extend forward of the lens board in much the same way.

The 35mm format has always been a bit of an odd duck when it comes to rules of thumb. Almost nothing is based on the actual film (or sensor) format; everything is based on an enlargement (traditionally an 8x10 print or an 8x loupe), since the 24mm by 36mm image is nearly useless in the real world. Except, it would seem, the word macro. There is some justification for that, in the sense that one needs to deal with "bellows draw" compensation in the same way. And for scientific and technical photometric applications (non-destructive testing being a prime example), actual 1:1 (as opposed to nominal "life-ish size") at the capture surface (especially on film) can be critical. But really, we're dealing with final images that are much closer to micro than macro. But the terminology is there, and has been for years, so for better or worse we're stuck with it.

The smaller the sensor format, the larger that "1:1" reproduction ratio will wind up being in a final image. "1:1" on an Canon APS-C sensor produces the equivalent (on an 8x10) of a 12.8:1 micro image, and if you wanted to press the "1:1" thing to the absurd, a "true macro" shot on a 1/2.3"-sensor compact camera would be a 33x enlargement (at 8x10) when all is said and done. At some point, we need to learn to let go of "true macro" and start talking about "35mm-equivalent macro" (or perhaps "APS-C-equivalent macro"). At least that will make some sort of sense when dealing with smaller camera formats.

Lenses of different focal lengths (and of different construction with the same focal length) will have different minimum focus distances with the same magnification of the subject at the focal plane. So if all you care about is big, your only concern is going to be the maximum reproduction ratio. If you are shooting flat subjects for reproduction (small documents — like stamps, say — or coins, or rock slices), that may indeed be all you need to care about. But life is rarely that simple, is it?

There are any number of reasons why you might want to get "the same" picture from different distances, and that's where the minimum focusing distance comes into play, even if the lenses (or cameras, in the case of compacts and bridges) can all make the subject the same size in the frame.

You may want a longer working distance. Again, there can be several reasons for this. You may need the distance because the thing you're shooting might not want you getting too close. You may want or need the light to be coming from an angle that would be impossible if the camera (or the front of the lens, or you, for that matter) were too close to the subject. You might want to minimize the amount of the background visible in the photo. Any of these reasons would push you in the direction of longer focal length lenses and a longer minimum focusing distance. But there are costs to longer lenses as well: the depth of field at any given aperture will be smaller (and at macro ranges, DoF is tiny with any lens), and any camera movement will be exaggerated. A hand-held life-size shot with a 400mm lens (both "life size" and "400mm" being 35mm-equivalent) is as much a matter of luck as skill, no matter how much time you've spent developing your camera-steadying skills.

There is also the little matter of perspective. Three-dimensional things will look bigger the closer you can get to them because their front bits will be proportionally much closer than their back bits. Again, with flat subjects, there's nothing to consider but the overall size, but if you want to turn that bug into a monster, make a tiny lizard look like a Permian-era nightmare, or make the elements on a printed circuit board look like urban architecture, you're going to want to be in really tight to exaggerate perspective. But then, being so close, you're going to have to deal with how you and the camera interact with the subject and the reduced lighting possibilities.

You may even find it necessary to "take it up a notch" by using a lens that allows not only high magnification, but perspective control and/or Scheimpflug corrections (a tilt/shift macro lens or a bellows with movements).

So what you need for "better macro photography" depends on the kinds of photographs you want to take. Any lens (or camera) that can take you into the appropriate magnification range will fullfil the most basic requirements (making the subject big), but magnification is only the starting point. You also need to balance the technical difficulties that macro photography is going to present against the artistic aims you have. And only you can make that decision.