Your position is phrased as a definition, so the assumption that it is true is tautological. If you assume that the word "object" can be defined by a set of atoms, and you assume there is a finite number of atoms (which is science's best guess, but not certain), then by definition there are a finite number of objects.
The question that does remain is "is this definition of 'object' useful." Is it useful enough to supercede all other definitions of "object" in the English language such that when you say "object" there is no ambiguity on the part of the listener.
That is a linguistics debate for you, which does not have an answer one way or another. However, I would suggest there is a linguistic argument to say "No, your reasoning is not correct: your definition declares that an electron is not an object." You are free to use your definition, but you must accept that there are things called "electrons" that others may think of as objects, but you define them to be "not objects."
This process can continue deeper and deeper if you so choose. It will get interesting if you reach the QM level, where the fundamental object is a waveform. At that point, the definition of "object" muddies greatly because we are pretty sure the waveforms start to blur together in some cases.
If I may suggest an interesting alternative definition...
The strange world of QM can be interesting if you define an object with respect to the way we think, rather than physics. Consider that nature really doesn't care if a table is one object, or five (a top and four legs). The way you define your objects matters when you start trying to predict what the table will do in certain situations (how will it break apart if I put too much weight on it?). This predictive value seems to be closely tied to our concept of what is an "object" and what is not; I consider this important.
With this definition, I can go through a a series of historical experiments that lead up to QM. You will see that, as we go along, the meaning of "object" shifts.
Physics purists: I'm certain I'm getting some terminology wrong. I am sorry in advance.
Consider a lightbeam. It is a meaningful enough thing that we might dare to call it an object (or maybe not... but I'm going to call it an object). If you have a lightbeam pouring into a room, the room is lit. If there is no lightbeam, the room is dark.
This is, of course, an approximation of reality. It is convenient to think of the lightbeam as a thing because we can use that to predict whether rooms are light or dark. This approximation is totally valid, up until the point where we start to strain it.
The double-slit experiment is a famous experiment. If you shine light on a slit, and observe the light on a surface behind the slit, you see a diffusion pattern: a bright patch in the middle which fades to darkness on either side. If you shine light on two slits, side by side, you see two diffusion patterns. However, if you get your hands on some monochromatic light (only one color/wavelength), strange behaviors occur. "Fringing" patterns appear, alternating bars of light and dark, much unexpected from our simple lightbeam model.
So we update the model, and have light travel as waves. We can do the math and show that the waves cancel out at the surface to generate these fringing patterns. We have had to modify our lightbeam definition a little: now each wavelength is broken out and has "phase" and "amplitude," but this is of minor concern. It's not that much more complicated, and it explains all we have seen.
Now consider a strange effect: the photoelectric effect. We observe that some materials, metals mostly, will emit electrons when lit up by light (which, of course, is how solar cells work). However, experiments show that it has some curious behaviors. If you use too low of a frequency of light, it doesn't emit an electron at all, no matter how powerful the light source is. Use a higher frequency of light, and it starts knocking electrons off, even with faint light sources. Curious.
The only way this makes sense is if light was quantized into little objects which knocked the electrons off like colliding billiard balls. We have to shift our definition of object... now there are billions of small objects, "photons," skittering about. High energy photons can kick an electron off of our metalic surface with ease. Low energy photons just lack the oomph. En masse, they sum to form wavelight behaviors, explaining the double-slit behavior. We've got a definition for an object that explains everything we've seen!
This is where it gets odd. Let's go back to our double-slit, now that we know light is quantized into photons. We can construct a light source that emits just one photon at a time. We can record where the photon hits the surface behind the slits, then send another photon. If we repeat this process and overlay all of the impact locations, we expect to see two bands, one behind each slit, because the photon has to go through one slit or the other.
Classical physics hangs its head in shame, because that is NOT what we see. We see fringing, just like before. Somehow the photon is traveling through both slits at the same time, and interfering with itself!
Quantum Mechanics can explain this behavior. QM states that photons are not particles, per se, but wave packets. They are waves that are localized in space, just like a staccato note played by a flute is localized in time. Within the note, there are vibrations, but they taper off as you approach the edges. If you model your "single-photon light source" like this, usually you will find its behavior to be approximately identical to the classical model. However, the double-slit setup amplifies the characteristics of this wave packet which classical modeling assumes "don't matter much." It shows that you should expect a fringe pattern, which you see.
Freaky, isn't it.
Note that, at each step, the "object" is whatever makes sense to help you predict behavior
In 99.99% of your life, you will not have to think about light as anything more than a beam.
A "lightbeam" is a useful object for you to work with. In a handful of odd cases, you will need to think of a light beam as something more than just one object. Perhaps you put a prism in the way of the light beam.
In 99.99999% of your life, you will not have to think about light as anything more than a wave.
However, perhaps you are interested in solar energy. You will have to think of photons as the fundamental "object" that light is made of.
In 99.99999 999% of your life, you will not have to think about light as anything more than photons.
However, if you play with the deep dark corners of physics, where new discoveries are found you wont have such luxuries. You will have to deal with light in the truest form we know: waveforms
In 0.00000 0001% of your life, you will have to think about light as a quantum waveform. If you deal with experiments such as the delayed-eraser-double-slit-experiment, you will defecate building materials when you see the results if you try to think in terms of classical mechanics instead of quantum waveforms. That being said, in all other cases, simpler objects are fine. You don't always need to use the most exacting model.
In 99.99999 99999 99% of your life, waveforms will be enough.
Are you a theoretical physicist? How cool would it be to find that the QM definition of light is insufficient because you ran an experiment that got different results than QM predicted.
Are you ready to redefine object once again?
Welcome back to the world of the sane. Objects are made of atoms again. Your sanity is intact. But you will never be the same, because you have seen the rabbit hole, and realized you may never truly comprehend how far it goes.
At each step of that journey, there was a thing that was worthy of being called an "object." Between each step, that definition was shaken until it crumbled. However, for most of your life, you are free to define an "object" to be something simple. In fact, it turns out that modeling the QM waveforms emitted by an incandescent lightbulb is virtually intractable. There's just too much math! If you use the best definition for an object made of light, you'll never get off the drafting paper. By choosing a less precise definition, you can make a useful product, like lightbulbs that we use every day!
These "less precise" defintions of "object" are useful because they help us understand the universe around us!