Plants most definitely have information processing. The entire process of 20th century computation reveals that you can get 'intelligence' from really stupid things (NAND gates) - conversing that we can have computation (and all that implies) from a variety of different substrates (neurons, mechanical relays, semi conductors, quantum bits, etc). People tend to get panicky and start delineating things so that they can restrict plants from 'having thought' but its all pretty silly.
Overall, seems likely to be selected against as a strategy for individual organisms.
To the original question, one obvious difficulty is considering how a mechanism encoding directionality in a chemical signal can evolve. It's certainly possible that these fungal networks are weakly directional, and it's also possible for a plant to evolve some sort of gradient sensing mechanism if its root network is sufficiently wide, but it -seems- unlikely. I would totally love to see if we can find an example of it though - it would be sooo cool.
Couldn't the population without the defector gene, eventually outcompete and extinguish the population where it can appear?
Communication between plants may not be cognitive, but it's still the transfer of information.
Chemical signalling of various kinds is very common in all kingdoms of life; examples in animals are hormones (circulated in the bloodstream for communication between organs), pheromones (circulated in the air for signalling between organisms), and neurotransmitters (emitted in the gap between synapses and dendrites for signalling activation of a neuron).