What if the constants are connected?
They are, but in the wrong way. From a popular text (and one that doesn't go too far overboard, in that e.g., it doesn't claim there is any evidence for god) :
"In most analyses of the fine-tuning of the force strengths and constants of nature, only one parameter is adjusted at a time (to make the problems more tractable). This would correspond to changing one dial at a time on our Universe- Creating Machine while leaving the other dials unchanged. Even taken individually, each of these examples of fine-tuning is impressive.
But in the real universe the values of all the constants and force strengths must be satisfied simultaneously to have a universe hospitable to life.
So for instance the strong nuclear force must be set to certain narrow limits for stars to produce carbon and oxygen in comparable amounts, for beryllium-8 to remain bound at least 10^-16 seconds, to keep the deuteron bound, to allow a minimum periodic table for life, to keep the light abundant isotopes stable, and to keep the di-proton unbound.
The range for each of these parameters is narrow. The range within which all of them are satisfied simultaneously is much smaller, like the bull’s-eye in the middle of an already tiny target. Add the required range for the weak force strength and the bull’s-eye becomes smaller still, and so on for the other forces. Add the specific requirements of simple life (water and carbon chemistries) and it becomes even smaller, and more so for advanced and then technological life.
Eventually, we will have a set of equations, each describing a different constraint on the laws of nature that allows them to permit life. (Determining this complete set of equations may just be the single most important goal of science. We’ll leave that as an exercise for the reader.)"
Gonzalez, G., & Richards, J. W. (2004).
The Privileged Planet: How Our Place in the Cosmos Is Designed for Discovery. Regnery Publishing.
When values are connected like these, we get a system of equations. Systems of equations are much more difficult to solve than single equations because they depend upon one another. You can't solve each one individually, you have to solve them all at once. This is a good portion of all science: finding solutions to systems of equations. The greater the tuning required for one parameter, the still greater tuning required for any dependent parameters (here I use "tune" and "parameter" in the more general sense to include e.g., actual models like climate models in which the key ingredient is the feedback parameter that is difficult to determine because it depends upon the microphysics of water vapor, cosmic rays, cloud dynamics, and so forth).
I think there is something to be said for the argument that positing a divine creator (or a multiverse, for that matter) just replaces the problem with another that now needs explaining. But I would prefer not to require an explanation at all, still less be left with the argument "well, your god solution doesn't work well either".