In the form you have phrased it, the statement is false or, at best, incomplete. In particular, the specific claim that
In a universe where the constant is a fraction higher, electrons would orbit closer to the atomic nucleus.
is essentially meaningless.
The problem is the assertion that, in a universe where X happened, the orbital radius would change, and it is problematic because the orbital radius is a dimensional quantity. What does the assertion really mean? it posits two universes,
- universe $A$, with fine-structure constant $\alpha_A$, and
- universe $B$, with fine-structure constant $\alpha_B\neq\alpha_A$,
but... how do you compare lengths between the two universes? You can't take a ruler from universe $A$ and use it to measure stuff on universe $B$, that's for sure.
You can try to 'port' lengths from one universe into another by specifying, say, $N$ times the wavelength of $X$ optical transition, or the speed of light times $M$ periods of the radiation from the $Y$ microwave resonance. However, what happens if you do this and the different lengths don't agree? Keep in mind that you're proposing changes in $\alpha$ so radical that the fundamentals of chemistry in your new universe would change considerably, so there might be no such thing as caesium or krypton in your new universe, or they would be altered beyond the point where you could reliably use them as standard rulers.
Instead, the way you make meaningful statements of this sort is to specify what it is you're keeping constant, or in other words to make predictions about the ratio of two different quantities (of the same physical dimension), which can be taken in-universe; you can then compare the answers (which are just numbers) from the two different universes.
Let me first stress that this is not a pointless insistence, and that there is a lot of active science research doing precisely this: the precision measurement of $\alpha$-dependent physical quantities, over several years, to see if they change. One good example is the paper
Frequency Ratio of Two Optical Clock Transitions in 171Yb+ and Constraints on the Time Variation of Fundamental Constants. R. M. Godun et al. Phys. Rev. Lett. 113, 210801 (2014), arXiv:1407.0164.
where it is crystal clear that the only observable which can be reliably used as a gauge for changes in $\alpha$ is the ratio between two commensurate quantities. In this specific case the two quantities being compared are the frequencies of optical radiation from a single trapped ion (because they can be measured to exquisite precision, and because there is strong theoretical evidence that those frequencies have a different dependence on $\alpha$), but the principle can be applied to the ratio of any two lengths, times, velocities - you name it.
I bring up this example to emphasize that the overall theme - asking what happens if $\alpha$ changes - is a perfectly reasonable question. However, you do need to use a more refined gauge on the effects of those changes.
OK, now let's bring it back to how you interpreted the control of $\alpha$ on the orbital radius, in the form
electrons would orbit closer to the atomic nucleus, which means atoms would never clump together.
The problem here is that the clumping together of atoms (i.e. the chemical bonds between them) is governed by the same physics that dictates the characteristic sizes of atoms. This is the quantum mechanical version of electrostatics, which is essentially ruled by the electron mass $m_e$, the Coulomb constant $e^2/4\pi\varepsilon_0$, and Planck's constant $\hbar$. None of these involve $\alpha$ directly (or, more precisely, it is possible to change $\alpha$ without changing any of these quantities), which means that the essentials of atomic physics and the fundamentals of chemistry remain unchanged.
This can be developed a bit further: in essence, saying that you care about the 'clumping' properties of atoms ultimately means that your lens is the dynamics that determines it, or in other words that you're willing to consider changes in $\alpha$ between universes $A$ and $B$, but these should be done by keeping $\hbar$, $m_e$ and $e^2/4\pi\varepsilon_0$ constant. This then tells you exactly what a change in $\alpha$ means, because we know that
$$
\alpha = \frac{e^2/4\pi\varepsilon_0}{\hbar c},
$$
and the only constant we can change now is the speed of light $c$. This is the real effect of $\alpha$ as far as atoms and molecules go, and indeed in atomic units $1/\alpha\approx 137$ is exactly the value of the speed of light.
As far as atoms and molecules go, then, changing $\alpha$ means that you keep the essentials of the situation unchanged, but you change the strength of relativistic effects, like spin-orbit coupling and other fine structure terms. (That name, of course, is not a coincidence.) These are small changes in the specific details of how atoms behave, which is precisely what the paper I talked about earlier is measuring.
However, this is only one perspective, and changing $\alpha$ will change other realms of physics differently. In deeply relativistic quantum regimes, for example, it makes much more sense to make $\hbar$ and $c$ the benchholders of your physics, and you switch from atomic units to natural units; here $\alpha$ is more meaningful as a measure of the Coulomb constant $e^2/4\pi\varepsilon_0$, which determines the strength of the electromagnetic interaction between two elementary charges.
This is a different perspective, one among many others; they are all equally valid, and for each a change in $\alpha$ will affect physics in different ways. For all of them you need to choose some dimensionless ratio as a hallmark that physics has changed - but the reference quantity in that ratio will be different, depending on what physics you're exploring, so you always need to say what it is you're keeping constant.
