Is the Boltzmann constant really that important?
We can understand all of this business if we visit the statistical mechanics notion of temperature, and then connect it to experimental realities.
Temperature is a Lagrange multiplier (and should have dimensions of energy)
First we consider the statistical mechanics way of defining temperature. Given a physical system with some degree of freedom $X$, denote the number of possible different states of that system when $X$ takes the value $x$ by the symbol $\Omega(x)$. From statistical considerations we can show that modestly large systems strongly tend to sit in states such that $\Omega(x)$ is maximized. In other words, to find the equilibrium state $x_\text{eq}$ of the system you would write $$ \left. \left( \frac{d\Omega}{dx} \right) \right|_{x_\text{eq}} = 0$$ and solve for $x_\text{eq}$. It's actually more convenient to work with $\ln \Omega$ so we'll do that from now on.
Now suppose we add the constraint that the system has a certain amount of energy $E_0$. Denote the energy of the system when $X$ has value $x$ by $E(x)$. In order to find the equilibrium value $x_\text{eq}$, we now have to maximize $\ln \Omega$ with respect to $x$, but keeping the constraint $E(x)=E_0$. The method of Lagrange multipliers is the famous mathematical tool used to solve such problems. One constructs the function $$\mathcal{L}(x) \equiv \ln \Omega(x) + t (E_0 - E(x))$$ and minimizes $\mathcal{L}$ with respect to $x$ and $t$. The parameter $t$ is the Lagrange multiplier; note that it has dimensions of inverse energy. The condition $\partial \mathcal{L} / \partial x = 0$ leads to $$t \equiv \frac{\partial \ln \Omega}{\partial x} \frac{\partial x}{\partial E} \implies t = \frac{\partial \ln \Omega}{\partial E} \, .$$ Now remember the thermodynamic relation $$\frac{1}{T} = \frac{\partial S}{\partial E} \, .$$ Since the entropy $S$ is defined as $S \equiv k_b \ln \Omega$ we see that the temperature is actually $$T = \frac{1}{k_b t} \, .$$ In other words, the thing we call temperature is just the (reciprocal of the) Lagrange multiplier which comes from having fixed energy when you try to maximize the entropy of a system, but multiplied by a constant $k_b$.
Logically, $k_b$ doesn't need to exist
If not for the $k_b$ then temperature would have dimensions of energy! You can see from the discussion above that $k_b$ is very much just an extra random constant that doesn't need to be there. Entropy could have been defined as a dimensionless quantity, i.e. $S \equiv \ln \Omega$ without the $k_b$ and everything would be fine. You'll notice in calculations that $k_b$ and $T$ almost always shows up together; it's no accident and it's basically because, as we said, $k_b$ is just a dummy factor which converts energy to temperature.
But then there's history :(
Folks figured out thermodynamics before statistical mechanics. In particular, we had thermometers. People measured the "hotness" of stuff by looking at the height of a liquid in a thermometer. The height of a thermometer reading was the definition of temperature; no relation to energy. Entropy was defined as heat transfer divided by temperature. Therefore, entropy has dimensions of $[\text{energy}] / [\text{temperature}]$.$^{[a]}$
We measured the temperatures $T$, pressures $P$, volumes $V$, and number of particles $N$ of some gasses and found that they always obeyed the ideal gas law$^{[b,c]}$
$$P V = N k_b T \, .$$
This law was known from experiment for a long time before Boltzmann realized that entropy is actually proportional to the logarithm of the number of available microstates, a dimensionless quantity. However, since entropy was already defined and had this funny temperature dimensions, he had to inject a dimensioned quantity for "backwards compatibility". He was the first to write $$ S = k_b \ln \Omega$$ and this equation is so important that it's on his tomb.
Connecting temperature and energy
In practice, it is actually rather difficult to measure temperature and energy in the same system over many orders of magnitude. I think that it's for this reason that we still have independent temperature and energy standards and units.
Summary
Boltzmann's constant is just a conversion between energy and a made-up dimension we call "temperature". Logically, temperature should have dimensions of energy and Boltzmann's constant is just a dummy that converts between the two for historical reasons. Boltzmann's constant contains no physical meaning whatsoever. Note that the value of $k_b$ isn't the real issue; values of constants depend on the units system you use. The important point is that, unlike the speed of light or the mass of the proton, $k_b$ doesn't refer to any unit-independent physical thing in Nature.
Temperature is the Langrange multiplier that comes from imposing fixed energy on the problem of maximizing entropy. As such, it logically has dimensions of energy.
Boltzmann's constant $k_b$ only exists because people defined temperature and entropy before they understood statistical mechanics.
You will always see $k_b$ and $T$ together because the only logically relevant parameter is $k_b T$, which has dimensions of energy.
Notes
$[a]$: Note that if temperature had dimensions of energy then under this definition entropy would have been dimensionless (as it "should" be).
$[b]$: Actually, this law was originally written as $PV = n R T$ where $n$ is the number of moles of a substance and $R$ is the ideal gas constant. That's not really important though because you can group Avogadro's number in with $R$ to get $k_b$. $R$ and $k_b$ have equivalent "status".
$[c]$: Note again how $k_b$ and $T$ show up together.
I think this question can be interpretted a few ways. I will frame the argument using the example of phase transitions.
1. Does temperature need to exist (or do we really need another constant)?:
No.
Let us consider what it means to have a phase transition. In the broadest terms, we are introducing energy into a system and approaching a critical point that either leads to long-range order (gas to solid) or disorder (solid to gas). We typically have defined these transitions to occur at a critical temperature. However, temperature is related to energy through the Boltzmann constant as
$$ E = k_bT $$
Thus, we could also define a transition energy instead of temperature which would remove the need for the Boltzmann constant. Therefore, I would say that we are more or less arguing that we could define the entire universe without a temperature like parameter, which is true.
2. Can we redefine temperature (or fixing our energy scale)?
Absolutely. However, we can do this for any fundamental constant and there is nothing special about Boltzmann. This is just trivial unit conversion.
3. What would happen if the relationship between the microscopic and macroscopic world were different? (or fixing the energy AND temperature scale)
In this interpretation, we would fix our temperature and energy scales, but change the relationship between the two. This would mean that the amount of energy required to heat or cool would be different. Thus, we are changing the amount of energy required to phase transition for example. We could still eliminate temperature and just use energy, but the amount of energy required would be fundamentally different.
In Conclusion
Temperature is an unnecessary variable as all physics can be simply transcribed in terms of energy. Thus, the Boltzmann constant could be removed. However, if we consider the temperature and energy scales fixed, changing the Boltzmann constant amounts to changing the energy required for many physical processes (i.e. phase transitions).
So how do we interpret the question?
The questions original statement specifically indicated that changing the Boltzmann constant would have no effect on the universe. Based on this, I interpret this question to be related to points 2 and 3. Since 2 applies to any constant as well, I think it's fair to assume the author meant 3.
If I'm mistaken and the author mean point 1, I believe the question should be reworded.
Perhaps the author is thinking that $k_B$ really serves as the rate of exchange between units that we use to measure energy and those we use to measure temperature (which are different more for historical reasons than anything else). From this point of view, if we doubled $k_B$, it would be the same as rescaling our definition of temperature, so things we now call 100 K are instead said to be at 50 K and so on. Of course, like any change of units this doesn't actually change anything physically.
This is fine, but it is not clear why the author thinks that changing the value of $c$ or another dimensionful constant is any different. The only type of constant whose absolute value clearly matters for the universe is some dimensionless parameter, like the fine structure constant $\alpha$ or the ratio of the proton to electron mass.