There are a bunch of criteria for employee vs gig, but the law isn't written prescriptively - so the person's work for you satisfies some of one and some of the other, then someone will have to decide how much control the contractor has over the working conditions (the greater the control, the more likely the classification of "contractor").
Generally, contractors should have expertise, use their own equipment, decide how the job should be done, and have a risk of losing money.
If you insist they use your lawnmower, that's a very quick employee classification (from what I've experienced). My gardener refuses to use any soil/mulch/whatever we own - likely for this reason. I've worked with real estate folks, and when they do a flip, they're very careful not to buy raw materials for the contractors lest some of them sue to be classified as an employee and collect benefits. They'll even use SW to set up an order with Home Depot, which emails the contractor's a simple way to place that order at HD and pick up, and then they'll ask the contractor to add the raw materials cost to the invoice. But they will not buy the materials for them.
Also, if I screw up my current job, I'll merely get fired. My employer cannot demand I repay wages (generally). If your situation with your contractor is comparable, he will be classified as an employee. They must take on risk of losing money if they negotiate poorly or screw up a job.
> It feels like instead we should just say this is a new thing, and create a new set of laws to regulate it in the way that we think is best.
I don't think anyone is disagreeing with this.