Does Schnorr's 2021 factoring method show that the RSA cryptosystem is not secure? Claus Peter Schnorr recently posted a 12-page factoring method by SVP algorithms. Is it correct?
It says that the algorithm factors integers $N \approx 2^{400}$ and $N \approx 2^{800}$ by $4.2 \cdot 10^{9}$ and $8.4 \cdot 10^{10}$ arithmetic operations.
Does this lead to the conclusion that the RSA cryptosystem is no longer secure in some sense?

Moderator note: a fourth version dated 2021-04-08 has appeared; this is pretty normal for eprint. It seems settled Claus Peter Schnorr (of Schnorr group and Schnorr signature fame) intends pre-publication of this paper, including abstract as on eprint.
No.
If the claim was true, then there would be an extremely simple way to prove it: $10^{10}$ arithmetic operations is nothing. There are tons of 800-bit factoring challenges available online. The author could just solve them and include the factorization in the submission; the lack of such a straightforward validation should be taken as empirical evidence that the claim is, as of today, unsubstantiated at best.
Leo Ducas, one of the top experts in lattice-based cryptography (and especially in its cryptanalysis) has implemented the March 3 version of the paper. The preliminary experimental evaluations seem to indicate that the method cannot outperform the state of the art (quoted from here):

This suggest that the approach may be sensible, but that not all short vectors give rise to factoring relations, and that obtaining a sufficient success rate requires much larger lattice dimension than claimed in [Sch21].


Personnal study (unfortunately, never written down cleanly) of this approach suggested me that this approach requires solving SVP in dimensions beyond reasonable, leading to a factorization algorithm much slower than the state of the art. My impression is that this is the consensus among experts having spent some time on it as well.

The corresponding Twitter thread is here.
Furthermore, this Twitter thread points to what seems to be a fatal mistake in Schnorr's paper.
(Warning: personal view) I'm also basing my conclusion on the fact that several top experts on SVP and CVP algorithms have looked at the paper and concluded that it is incorrect (I cannot provide names, since it was in the context of anonymous reviews). Of course, the latter should not be treated as clear evidence, since I'm not providing proof of this statement - please treat it simply as it is, a claim I'm making.
(This statement refers to the version of the paper which was initially uploaded, and whose eprint extracted abstract  contained a claim that RSA was &quot;destroyed&quot;, together with the sample running times given in OP's question. Schnorr himself still claimed, after being asked by mail about the paper, that the latest version breaks RSA - and so does its abstract; with respect to this claim and given the lack of solved RSA challenge, I stand by my statement that it should be regarded as essentially unsubstantiated).
Among the potential issues (again to be treated with care, as pointers to help people willing to look further into where the paper might fail):

The proof of (5.8) does only show the existence of many smooth triples, but says nothing about the probability of successfully finding factoring relations.
The paper relies on the Schnorr-Hörner pruning strategy, which is known to be flawed.
No justification is given for the cost indicated for (3.2)