It's technically true that the classical SM Lagrangian is scale-invariant for zero Higgs mass, but this fact is not useful for model building.
First it's worth reiterating the point of naturalness. The goal is to specify a simple UV model and a simple distribution over bare parameters, so that the likelihood of attaining physical observables like those we actually see is high. (This is quite different, but equivalent to most other statements of naturalness; for discussion of why see my dissertation or this answer. One can, rather loosely, express the goal of all science in this way -- you are always searching for a simple model that is likely to produce what you actually see in the world. We care about bare parameters because that is what higher energy theorists, e.g. string theorists, will hand to us -- their output is our input.)
Implicit in most naturalness arguments is the idea of a Wilsonian cutoff. Your bare parameters are the coefficients in a Lagrangian with a high cutoff $\Lambda$, while physical observables are directly related to the coefficients in the effective Lagrangian once you flow to a low cutoff.
In this thinking, the Wilsonian cutoff itself strongly breaks scale invariance. For example, Higgs loops will go as $\Lambda^2$. Scale invariance is not restored when the Higgs mass goes to zero.
The benefit of the Wilsonian cutoff is that it acts as a generic stand-in for any physical effects that come in at scale $\Lambda$. In other words, some currently unknown physics can be in effect above scale $\Lambda$, which could be new particles, a compositeness scale, a lattice (the original inspiration), or even something which isn't a quantum field theory (such as string theory). The standard practice with naturalness just black-boxes this, because at the end of the day such a theory can be expressed as some EFT with cutoff $\Lambda$, and asks which classes of EFTs could yield the physics we observe.
Now, your idea is known in the literature as "finite naturalness" or "physical naturalness". For it to have any chance of working, you need a regulator that is compatible with scale-invariance, of which the best candidate is dimensional regularization. But the problem is that, while the Wilsonian cutoff was wonderfully generic, dimensional regularization is incredibly fragile. For example, the addition of any new heavy particles of mass $M$ (that are not very weakly coupled) will introduce another breaking of scale-invariance. Your Higgs loops will contribute $M^2$ again, reintroducing the problem.
That is, if your theory is natural with a Wilsonian cutoff, it'll stay natural no matter what is lurking above that cutoff -- it allows anything. But if your theory satisfies finite naturalness with dimensional regularization, nothing can appear at high energies. That's a totally different constraint, and it means that for finite naturalness to work, you have to explain neutrino masses, dark matter, the strong CP problem, and even quantum gravity, all without heavy particles. That is, as just one component of the first step, you'll have to cook up your own alternative to string theory!
While there has been work done in this direction, such as Agravity, which solves the nonrenormalizabilty of effective quantum gravity by just imposing dimensional regularization "all the way up", finite naturalness is generally regarded as a long-shot approach which is more of a career-long research programme than a one-line fix. Of course nature could work this way, but it's not as easy as it looks.
