But if you then keep mowing the lawn regularly, those seeds won't be able to compete with the grass.

Unless you mow your grass too low. Always assume the old rule of "your grass reaches just as far underground as it reaches up in the air" still holds.

Also if you mow your grass drastically shorter or you let it grow for a long time before mowing, do not fail to fertilize it from above right or soon after, start aggressively plucking the leaves of weeds (or other selective methods of fighting them) for a few weeks and (optimally, but highly recommended) verticulate it no sooner than 1 week after cutting. Also time it well to grant your lawn at least 3 weeks of ideal growing weather and climate (It won't die because of a week or two of awful weather, but you'll have A LOT more work fighting weeds ahead of yourself).

Why wouldn't they be able to compete?