It may help to focus on how pressure centers aren't initially about rotation.
Picture you are air... you start moving towards a new cyclone (because of a lowered pressure). Straight inwards...
Then you start to turn to the right because of Coriolis (in the NH).
That results in only the right diagram (try adding the arrows in from other directions)
The thing is, even if you managed to get a clockwise circulation like your left diagram... its basic forces don't suggest it can continue to work. Because the two forces of note typically directing the air (horizontally) in pressure systems... both point towards the low pressure if the low is rotating clockwise; coriolis (turning to the right, NH) and the pressure gradient both pull the wind directly inward towards the center. The circulation would cease.
Circulation is all about balance.
It is actually indeed possible to get a clockwise low pressure circulation. But, given those pair of forces it would need to counteract, you need something relatively hefty to make it happen. Probably the most viable method to start up a clockwise low capable of persisting is by the tilting of vertical rotation (i.e. wind shear), which will bring a counterclockwise area of rotation north (NH) of an updraft in a thunderstorm. (See Figure 3 from Dynamics Of Tornadic Thunderstorms, Klemp, 1987) So this means there are clockwise rotating mesocyclones.
From there, you can have enough rotation speed that centrifugal force can begin to be a bigger factor, which does point outwards [that's the balance that keeps satellites rotating around the planet... gravity pulls them in... centrifugal force pushes them out... result is they can maintain their height]. And that can allow them to persist, and even tighten up... leading to anticyclonic tornadoes, which are indeed low pressures that are rotating clockwise.
There are other options for forced clockwise circulation, like a downdraft/rear inflow jet surging, leading to a diagram like this one from SPC's bow echo education:
(Note the anticyclonic on the bottom)
However, these tend to weaken more/aren't in a favorable spot for continuation.
But it's only that such clockwise rotation can't develop "on its own" without those additional forces/motions circumstances helping, and then needs just the right balance to continue (counterclockwise rotation does need just the right balance too, but those two opposing forces mentioned [PGF+Coriolis] tend to really help the balance be much more stable in the face of disturbance).
For a little more discussion of the balancing tendency in a cyclonic (counterclockwise in NH) low pressure, see also this question here.