Your basic misunderstanding is that $|\Sigma(k)G_0(k)|>1$, which is not true and basically three principles are used at least to make sure that $|\Sigma(k)G_0(k)|<1$. First is very basic which is the renormalizability of the theory and second is local nature of counterterms. A third ingredient is not needed except in case of scalar particles which is termed as "naturalness" in the standard model.
The mass correction term $\Sigma(k)$ is a sum of infinitely many graphs contributing to the self energy of the particle which in general will have a linear divergence (like self-energy of electron) or quadratic divergence (like self energy of photon) in high-energy limit. 28
In case of particles other than spin zero different regularization schemes can be used to make this behaviour only logarithmically divergent and a redefinition of mass parameter ( renormalization) absorbs this divergence.
This basically tells you that $\Sigma(k)$ behaves as $ln(k/m)$ with k can be replaced by a scale $\Lambda$ and you will have $|\Sigma(k)G_0(k)|$ behaviour as $ln(k)/k^2$ as $$G_0(k)=\frac{1}{k^2+m^2}$$
Now $ln(k)/k^2$ is always less than $1$ (as $ln(k)-k^2$ is always negative).
In case of scalar particles however you have $\Sigma(k)$ behaving as badly as quadratically divergent (not enough symmetries in the theory), for which you need to have a lot of cancellation involving $\Lambda^2$ term to smoothen out the resulting integrals to not diverge as badly as quadratic. There is however no reason to expect that such cancellations should "naturally" occur as a result you need to fine-tune the theory to obtain a proper cancellation in which you can have $m^2_{physical}<<\Lambda$. As a result of these cancellations, you will have a soften up behaviour of $\Sigma(k)$ which will not diverge as badly as $\Lambda^2$ and you will have $|\Sigma(k)G_0(k)|<1$ as required.
Fine-tuning problem however is not a physical solution and hence must be supplemented with extra symmetry of theory. A case of which is supersymmetry which supplies the symmetry behaviour of spin 1/2 superpartner to make the divergence logarithamic but that is another question.
I hope this explain all your concerns and try to get a grab of good qft book like <PERSON>'s qft. 976