The 'general' TSP (unless P=NP) does not admit a worst-case constant-factor approximation, at least not on classical machines (we're still working on the quantum PCP theorem).

So while I'm not immediately familiar with the approximation algorithm you're suggesting (unless it is Christofides[1]), it seems unlikely that it would produce a good approximation for the TSP. It might still perform well in the average-case over random instances. I'll admit I haven't delved deeply into that about the TSP, but I don't believe there are currently any favorable results. There are certainly no Overlap Gap Property related results about the TSP. If I wanted to prove something in the average case about the TSP, I'd definitely be starting there.

[1]: If you were thinking about Christofides: Christofides is indeed a 3/2-approximation, but not for the general TSP. Instead, it approximates a problem like the TSP, known as the Metric TSP. The Metric TSP introduces additional symmetries that make it much, much easier to approximate. As such, Christofides is an approximation algorithm for an approximate version of the problem, which I think is pretty neat. If I'm remembering correctly, the inapproximability results for the Metric TSP are rather favorable.