What are your plans for the area? sod, seed, dog run?

Since there is no grass/lawn to speak of at this time, now is an excellent time to dig up and remove as much of the netting as you possibly can. Plastic netting is typically only photodegradable at best, requiring exposure to sunlight and many years to break down. And it doesn't really disintegrate or dissolve but simply breaks down into microplastics. And who knows how long they will remain present or how toxic they may prove to be.

Dogs are always going to be hard on a lawn, specially big dogs. If you want a lawn, you might think about partioning off an area just for the dogs and plant the lawn elsewhere. Wood chips are often used at dog parks because they are usually available for free, are durable and keep the mud and destruction to a minimum.