Kyle, there are a couple of things going on in this context.
First, the concept of the "seed bank" - every soil has one, it is all the ungerminated yet viable seeds that are lying dormant in the soil, waiting for the conditions that favor them. Seeds can sit around in this state for long periods, years for some. So, let's say you mow everything down before it seeds. Your next round will come from your seed bank in the soil, plus whatever migrates onto your property. Lots of vectors for migration of seeds onto your property; animals will carry them in, the wind will blow them on,
water running off of an adjacent property will carry some, so there is always going to be some question of new seed coming in.
Then there's the whole issue of how plants propagate. Seeds are only one mechanism. You get rhyzomes and runners, bulbs divide, you get layering (where a branch touches the ground and sets
roots, making its own new plant) - plants are adapted to reproducing despite all kinds of obstacles.
So you mow it all down before it seeds - sure about that? different plants are setting seeds at different times, some plants are so low growing you probably are not going to mow their seed heads off. But even if you do mow so that none of your plants go to seed - your plants will respond, at least some portion of them, by reproducing through other means. So you will keep having plants. You may be selecting for exactly the plant you want to eliminate, depending upon whether or not it is one that has other means of multiplying and is favored by you chopping down its competitors. You may actually be able to beat it back, if it is seed dependent.
So, let's say you have a problem plant that propagates by seed and only seed. You cut it down and next year all of its sees from two years ago that are still in the ground take their shot. It could be years before you have used up the seed bank.
But, will mowing before things go to seed lead to a desert? Not very likely. It may lead to an area totally dominated by plants that reproduce without seeds. It may lead to plants developing that grow really, really low fruiting bodies that hide below your mowing level, along with those that do not need to set seed at all.
Ever notice how some lawns have tall, upright dandelion flowers and some have little buttons right at ground level? You can tell who mows, how often and how high