Your definition is exactly that of magnetic field strength, H, on the old cgs system. The oersted was its unit, a magnetic field strength of 1 dyne per unit pole. [A unit pole was such that if two unit poles were placed 1 cm apart in vacuo, there would be a force of 1 dyne between them.]
The study of electromagnetism was built on this foundation, and produced the edifice of electromagnetic theory that we have today – even though we may now choose to define things differently. So one answer to your question would be that there's nothing wrong with the 'per unit pole' definition. It has delivered the goods.
Another view is that it's odd to use the non-existent monopole as the basis for electromagnetism. In fact the subject used to be taught using long ball-ended (dumb-bell shaped) magnets. The ball at the 'North' (or South) end produced a magnetic field in the surrounding air radially outwards (or inwards) varying with an inverse square law. So the balls seemed to be behaving as monopoles. But there's a very important caveat… The net magnetic flux from the North pole ball (unlike from a North monopole) is zero! This is because as much flux approaches the ball through the 'bar' of the magnet as leaves through the air, and as much flux leaves the South pole ball through the bar of the magnet as enters through the air. One of the consequences is that we have to be especially careful studying magnetic materials using this approach. Arguably an approach based on Ampère's current loops or the corresponding quantum concept is less contrived.