Although you haven't explicit said it, I get the impression that you are wondering if the force is somehow split between providing rotational vs. linear acceleration. That does not happen.
It's a both/and situation, not an either/or situation. A force can cause both accelerations simultaneously.
The linear will be (instantaneously)
$$\vec{a}=\frac{F_{net}}{m}$$
and $$\vec{\alpha}=\frac{\Sigma (\vec{R}\times\vec{F})}{\mathcal{I}},$$
where $\vec{R}$ is the vector pointing from the point of rotational interest to the application point of each force in the sum. For your problem, you only have one force. The difficulty will become how does that force change over time, and how do you describe it mathmetically.
But the force isn't split into different roles. It plays both roles.
I do not think it is mass that "CAUSES" inertia. Rather mass is DEFINED to be the property of an object which gives it inertia
For example, imagine that you were a theoretical observer who did not know of the concept of mass. There were just 3 objects in front of you. You applied $10 \mathrm{N}$ to each object and observed that the different objects accelerated at a different rates. One accelerated at $1 \mathrm{m}/\mathrm{s}^2$. Others at $2\mathrm{m}/\mathrm{s}^2$ and $4\mathrm{m}/\mathrm{s}^2$ respectively. So, you can conclude that the different objects have some property that determines how much they accelerate under a given force.
You would also notice that there was a linear relationship between the force on a body and its acceleration.
Now, you can calculate what would be the value of this property. And you can see that the body that accelerated at $1 \mathrm{m}/\mathrm{s}^2$ would have a higher value of this property than the one that accelerated at $4 \mathrm{m}/\mathrm{s}^2$.
The theoretical observer can see that this property is directly related to what people call "heavier" and "lighter" objects. So, it is safe to conclude that lighter objects accelerate faster than heavier objects.
To summarise, the reason lighter objects accelerate faster than heavier objects is because that is how we have defined the terms "lighter" and "heavier".
Mass "causes" inertia, because that is how we have defined "mass".
If you want to further ask, why objects with different values of this "mass" property accelerate differently, then you are asking a more fundamental 'why' question. And like most fundamental why questions in physics, it boils down to , because those are the laws of physics in our universe. And if you want to ask why are the laws of physics like that, the answer is " because of the initial conditions at the Big Bang. And researchers are working to figure this out "
EDIT : As some comments have pointed out, there is also the quantum mechanics aspect of this.
All particles get their inertia or the "tendency to resist motion under a force", through the way the fundamental particles interact with the Higgs field.
To understand why massive objects have inertia, we need to understand where the mass of objects comes from.
We know that all massive objects are made up of atoms. So, atoms give objects their mass. But where do the atoms get their mass from?
Atoms are made up of protons and neutrons (and electrons but they have negligible mass). So, protons and neutrons give atoms their mass. But where do protons and neutrons get there mass from ?
According to the standard model, protons and neutrons are made of quarks. So, quarks should be giving atoms their mass, right ? WRONG.
Quarks have substantially smaller and lighter mass than the protons and neutrons that they comprise. It is estimated that the masses of the quarks, derived through their interaction with the Higgs field, account for only about 1% of the mass of a proton, for example.
So, 99% of the mass of a proton is not to be found in its constituent quarks. Rather, it resides in the massless gluons that bind together the quarks inside the proton. These gluons are the carriers of the strong nuclear force, that pass between the quarks and bind them together inside the proton. These gluons are "massless" but their interaction with the Higgs field is what we experience as mass.
So, at a quantum level , the Higgs field ‘drags’ on the gluons , as though the particle were moving through molasses, (Another analogy used is moving through a crowded dance floor) . In other words, the energy of this interaction is manifested as a resistance to acceleration. These interactions slow the particles down, giving rise to inertia which we interpret as mass.
Best Answer
A few simplifying assumptions:
OK, so now, let's analyze our idealized bicycle. We're going to have the entire $m$ of each of the two wheels concentrated at the radius $R$ of the tires. The cyclist and bicycle will have a mass $M$. The cycle moves forward when the cyclist provides a torque $\tau$ to the wheel, which rolls without slipping over the ground, with the no-slip conditions $v=R\omega$ and $a=\alpha R$ requiring a forward frictional force $F_{fr}$ on the bike.
Rotationally, with the tire, we have:
$$\begin{align*} I\alpha &= \tau - F_{fr}R\\ mR^{2} \left(\frac{a}{R}\right)&=\tau-F_{fr}R\\ a&=\frac{\tau}{mR} - \frac{F_{fr}}{m} \end{align*}$$
Which would be great for predicting the acceleration of the bike, if we knew the magnitude of $F_{fr}$, which we don't.
But, we can also look at Newton's second law on the bike, which doesn't care about the torque at all. There, we have (the factor of two comes from having two tires):
$$\begin{align*} (M+2m)a&=2F_{fr}\\ F_{fr}&=\frac{1}{2}(M+2m)a \end{align*}$$
Substituting this into our first equation, we get:
$$\begin{align*} a&=\frac{\tau}{mR}-\frac{1}{m}\frac{(M+2m)a}{2}\\ a\left(1+\frac{M}{2m} +1\right)&=\frac{\tau}{mR}\\ a\left(\frac{4m+M}{2m}\right)&=\frac{\tau}{mR}\\ a&=\frac{2\tau}{R(4m+M)} \end{align*}$$
So, now, let's assume a 75 kg cyclist/cycle combo and a 1 kg wheel, and a 0.5 m radius for our wheel. This gives $a=0.0506 \tau$. Increasing the mass of the cyclist by 1 kg results in the acceleration decreasing to $a=0.0500 \tau$. Increasing the mass of the wheels by 0.5 kg each results in the acceleration decreasing to $a=0.0494$, or roughly double the effect of adding that mass to the rider/frame.
This result, e.g. an ounce of weight at the rims is like adding two ounces of frame weight is true regardless of the mass of the cyclist/cycle, wheel radius or rider torque. To see this, note that $$\begin{align*} \frac{da}{dm} &=\frac{-8\tau}{R(4m+M)^2}\\ \frac{da}{dM} &=\frac{-2\tau}{R(4m+M)^2} \end{align*}$$ Adding a small amount of mass $\delta m$ to the frame changes the acceleration by $\delta m \frac{da}{dM}$ while adding half this amount to each of the wheels changes the acceleration by $\frac{1}{2} \delta m \frac{da}{dm}$. The ratio of acceleration changes is $$\frac{\frac{1}{2} \frac{da}{dm}}{\frac{da}{dM}} = 2$$ regardless of the other parameter values. It's not hard to see that this result is true for unicycles and trikes as well (i.e. doesn't depend on the number of wheels on the cycle).