There aren't any "advantages" per se. I putt best with heel shafted putters with significant toe hang, so it would be a huge disadvantage for me if I had to putt with a center shaft. It's just a question of whether it fits your stroke style or not. IMO, there's not a great way to know one way or the other without trying one, and with any putter, it's got to feel right, and you need to be able to hit it where you're aiming.
For example, because I putt better with a heavier toed putter (meaning I hit those where I'm aiming with my normal stroke), so if you handed me a true center shafted putter (i.e. face balanced), and I putted with my normal stroke, I would consistently miss left because the toe is "lighter," and the face closes much faster.
FYI, this is from an older thread. I saved it for my general knowledge of center-shafted putters vs. others. I'm pretty sure the author posts on GolfWRX quite regularly.
Here's a brief "center-shafted putter" theory by Putting Guru, Geoff Mangum:Center Shafted vs. Heel Shafted: Why should you use a center-shafted model ????
The key word here is "tastes." A taste is a habit. The Chinese eat dog meat. Some folks in South America eat bug larvae. Some folks in southeast Asia eat monkey brains. Historically, putters were usually heel-shafted, mostly because other clubs are heel-shafted and also because the Brits (R&A) banned center-shafted putters for about 50 years. The humorous thing is that center-shafted putters now have such success in the market, that even some designers who previously swore never to make a center-shafted putter have jumped on the bandwagon after abandoning all self-respect and principle. If you use a heel-shafted putter, the tool will teach you the "habit" of success with it, or else you will discard it. So if you keep the heel-shafted putter, its inherent physics has trained you into a habit. The habit consists partially of habitual patterns of perception and partially of patterns of movement and feelings. For example, a heel-shafted putter looks and feels a bit more like swinging a bat around your stance, with certain implications for how you perceive the stroke and ball impact, and how you expect things to look and feel in the movement. The instincts rely upon these habits.
If you use a center-shafted putter, the tool will teach you the "habit" of success with it, or else you will discard it. A center-shafted putter swings more vertically up thru a ball, unlike a baseball-style swing with a heel-shafted putter sideways thru a ball. This implies a different set of visual and kinesthetic "habits" in using the center-shafted putter.
You can use a heel-shafted putter with the same habits appropriate to a center-shafted putter if you add a trick or two, and vice versa. (Many if not most belly putters are center-shafted, yet swung around the stance more like a bat than a pendulum.) The bottom-line question is which is better, and what is required for you to unlearn inappropriate habits and to learn new, appropriate habits if you switch?
In my view the center-shafted putter is better in general than heel-shafted putters, because heel-shafted putters have physics in them that promote action of the putter separate and apart from what the golfer is deliberately intending. In particular, the typical hoseling and balancing of a heel-shafted putter promotes so-called "toe flow." This is an EXTRA opening of the putter going back that may or may not be matched coming forward. The physics of "toe-flow" is the added inertia in the stroke of the toe about the axis of the hosel. This physics opens the toe going back, and tends to KEEP the toe open coming thru impact. This doesn't make a lot of sense.
The golfer who gets trained by his heel-shafted putter has to learn how to manipulate the physics of the putter correctly. This obviously can be accomplished over time -- witness Ben Crenshaw. But why engage in a battle that is not compelled? Just don't get a heel-shafted putter. The center-shafted, face-balanced (or reality balanced) putter doesn't have these same tendencies from physics. Then the golfer's task in getting trained by the tool is a little easier and simpler.
Heel-shafted putters are an historical accident that some people seek to justify with a bogus rationale. What the physics really does is make the putter designer an unwitting partner in every stroke: you do this and the designer adds that. I prefer to putt alone.
So when you switch from heel-shafted to center-shafted, you get a slight unburdening, but you are temporarily stuck with old heel-shafted habits. It takes a while to learn the new "look and feel" of the center-shafted putter, and thus "acquire the taste" for the tool.
A compromise is a putter with the actual hoseling towards the heel, but for which the shaft AIMS at the center. These putters are face balanced, and appear to be heel-shafted, but are really center-shafted with heel-hoseling. This would be a good transition putter, allowing you to ease into the tool's training of you without a lot of contrast with old habits.
Keep your tastes -- just get rid of unhelpful physics.
Also CS putters are supposed to be better for left eye dominant golfers, like me. Dont know if it helps, i just like the look!