It may not be relevant to your situation but grass is the one plant that I aggressively weed out of my
garden beds. Here they grow by above and below ground runners, as well as
seed. They can quickly cover the whole garden bed and make it nearly impossible to plant new plants. Consider that a big part of the reason they are so commonly used for lawns is that they grow vigorously enough to keep out other plants (i.e. weeds) and withstand regular trampling. Much more than half of all my
gardening efforts are involved in just controlling the grass.
That's not to say you're on the wrong track. It sounds like what you seek is an effective living mulch and there are many of those. Clover is probably the most well known. There's a member on here (Bryant RedHawk) who has frequently laid out the process of using a combination of plants including living mulch to successfully do large scale farming.
There's also the option of using seasonal annuals as living mulch in rotation with other plants. I've had good luck with sweet potato vines keeping the grass back (though I'm still working on how to grow other plants with the sweet potatoes) And one of the reasons behind squash in the three sisters garden is that it serves as living mulch for the other plants. Popular in the organic
gardening in my region is vetch as a nitrogen fixing winter cover crop and buckwheat as a fast growing builder of organic matter and pollinator attractant in warm weather.
I'm experimenting for the first time this year with a densely planted row of swiss chard to try and block the grass. It hasn't sprouted yet, but I've planted a block of swiss chard at the end of my garden where the rows meet the
lawn.