Because - as I have attempted to explain - time dilation is a relationship between distant clocks, whereas a field assigns a particular object (a tensor of spinor of any rank) to each local event in spacetime. You cannot point to an event in spacetime and say “I am going to assign time dilation factor X to this event”, without any further qualification - this does not make any physical sense.
The most fundamental entity in GR (and the solution to the Einstein field equations) is the metric tensor field - it assigns a metric tensor to each event in spacetime. To put it in the simplest possible terms, the metric tensor field allows you to quantify how each event in spacetime is related to all other events - both in spatial terms, and in terms of time. It does so by defining a mathematically precise relationship between neighbouring events, so that, by integrating along curves, you can calculate relationships between more distant events, e.g. the length of a world line connecting them.
Time dilation in GR is a geometric property of world lines, in that it is the ratio between the lengths of world lines between the same events - the total time a clock accumulates between two given events is equivalent to the geometric length of the world line traced out by that clock. And how long that world line will be depends on the geometry of the spacetime it is in, and what kind of world line it is.
Take for example a rotating spherical body, such as a planet. If you let a test clock orbit the planet once in its direction of rotation, starting and finishing at some point P, then that orbit will take a total time T1. If you now start at the same spot P, but orbit in the opposite direction (counter the planet’s direction of rotation, but along the same orbit, with all other initial and boundary conditions remaining equal), you will get some orbital time T2, which will be ever so slightly different. That’s because, even though you start at the same point P, and traverse the same spatial distance along the same orbit, the geometry of spacetime is such that the lengths of the two world lines will differ. The ratio between these two geometric lengths is one example of gravitational time dilation - the value of that ratio depends on where the point P is, the initial and boundary conditions of the clock kinematics, and the global geometry of the underlying spacetime. How would you capture all this by assigning a single value to point P, as you seem to want to do with your “time dilation field” idea?
Again, on closer consideration, in order to capture all relevant degrees of freedom so that all aspects of gravity can be correctly modelled, independently of the precise circumstances, at least a rank-2 tensor field is necessary. That’s what GR does.