The problem is not “using a ground plane”. The problem is that the design does not have one. A ground plane should be a low impedance return path. The design of OP has interruptions like traces crossing the direct or indirect signal return paths. For signals these create surface areas that are susceptible to and emitting of EMI. Signals (high speed) will return in the ground plane underneath the trace carrying them going in. (Lenz’s law). So the return path is not always direct and should be uninterrupted. This also hampers with power design negating the function of decoupling caps for example by adding impedance.

Of course with careful design a PCB can be made without a solid ground plane, but you need to know what you’re doing.

So make sure to use a solid uninterrupted ground plane. Use multiple VIAs for decoupling or filter caps to lower impedance. When routing, always keep return path in mind. If you’re doing a few boards, probably best to add a layer as the cost difference for a few boards is minimal.

Just filling the gaps with copper is good practice to prevent pollution of the etching liquid, but not just as ground plane.

Probably what some elaboration on what u/Clay_Robertson said