I'm revisiting Calculus and learning more about infinitesimals. As I concluded one of the reasons they introduced infinitesimals is because of "rounding real numbers problem" (read third paragraph). For example, $2+10^{-(10^{10^{10}})}$ is close to $2$, but still it is equal $2+10^{-(10^{10^{10}})}$ not $2$. In other words we can say $\displaystyle2+10^{-(10^{10^{10}})}{\simeq2}$ but not $\displaystyle2+10^{-(10^{10^{10}})}=2$.
In contrast, if $\mathrm{d}x$ is infinitesimal or, by definition, number which is larger than zero and less than any real number, then we can neglect that term and say $\displaystyle2+\mathrm{d}x=2$. Let's imagine for a moment that $\mathrm{d}x=10^{-(10^{10^{10}})}$. Isn't it sufficiently good to use such number to be neglected, so we can obtain "slope of a curve at a point"? In other words if $f(x)=x^2$ and $\mathrm{d}x=10^{-(10^{10^{10}})}$ then $\displaystyle\frac{f(x_0+\mathrm{d}x)-f(x_0)}{\mathrm{d}x}=2\cdot x_0+\mathrm{d}x\simeq2\cdot x_0$. We rounded number $2\cdot x_0+\mathrm{d}x$ to $2\cdot x_0$.
This might sound weird in context but what kind of effects does rounding real numbers (such as number $\pi$ to $3.14$) have on whole idea of real number system? If it doesn't have any, then why do we introduce infinitesimals at all? We could use any sufficiently good $\mathrm{d}x$ where $\mathrm{d}x$ is real number (such as $\mathrm{d}x=10^{-(10^{10^{10}})}$) and get slope of a curve at each point using only real numbers and not including hyperreals! My amateur opinion is: if you could round every number then there will be "gaps" on real number line (and I know in layman terms that real numbers represent continuous number line). Hence we introduce infinitesimals as extension to real numbers where infinitesimals are not in a domain of real numbers.
This is at least in part a philosophical question as well as a mathematical one.
Mathematicians have worked for centuries to build a logical foundation for the mathematical object we refer to as "the real line". The traditional view today is that the way to deal with small numbers is to use "epsilon-delta" proofs for limits. These replace the idea of "infinitely small" by "arbitrarily small" - infinitely many inequalities instead of single equalities involving infinitesimal numbers. There's a way to do it with actual infinitesimals you hint at in your last sentence. It's called "nonstandard analysis".
In the meanwhile, mathematicians starting as early as Archimedes and continuing through Leibniz and Newton to the present day have developed calculus for both theoretical and practical purposes even while the foundations were under construction. Sometimes that's referred to as "sufficient unto the day is the rigor thereof."
When actual numerical calculations are required, whether by Gauss computing orbits or today in computers, mathematicians carefully analyze the rounding errors that can occur when finite decimals are used to approximate theoretical "real numbers". Sometimes that's a bit like choosing a very small value for $dx$.
Related: Why can't the second fundamental theorem of calculus be proved in just two lines?