To be fair, if I recall it's only in the last 15-20 years that the Altaic theory seems to have gone from 'speculative' to 'not at all likely.' Those of us who studied this stuff 20 years ago could easily spout off this 'might-be fact' if we didn't bother to verify first.
I guess at this point similarities between the two languages are assumed to be a result of regional influence?
Many mainstream linguists have been against the hypothesis since the 1950s, but there are still plenty of linguists who contend that the Turkic, Tungusic and Mongolian families form a family; a smaller number have arguments in support of the stronger claim that the Koreanic, Ainu and Japanese–Ryukyuan languages are connected as well. See eg: [0] Vovin set out to demonstrate that the latter, stronger claim was true and ended up writing this: [1]
Arguably, the biggest blow struck to the hypothesis in the last 15-20 years came when Vovin, previously a supporter of the hypothesis, wrote "The end of the Altaic controversy"[2]. Among the more notable arguments in it is that the non-traditional methods[3] used to argue that Altaic is a family are rejected by traditional comparative linguistics because they overweight lexical comparisons.
This was a rather prescient point, as some of the methods that have developed since then that have been used to argue in favor of the Altaic hypothesis suffer from this problem quite badly; this is perhaps especially true of the Bayesian phylogenetic inference, sometimes called Bayesian phylolinguistics when applied to historical linguistics. With this technique, used in eg [0], it is pretty hard to argue that this method wouldn't overweight cognates and loan words gained through contact unless you specifically control for that, which will cause you to end up erroneously marking sprachbunds as true families.
See [4] for another usage of the technique, this time to support the Dravidian family, which is already very very well support by traditional comparative methods.
All of this ultimately points to what I think is the real answer to the issue; outside of some arguments about archaeology that I don't have enough background to evaluate [5], most of the shared bits (shared pronouns, lexical stuff) between the languages here exist because of contact.
The languages here, if they belong grouped at all, should be grouped as a sprachbund, not as a family. The possibility that there is a true family with "micro-Altaic" is small, but I think it is still a much greater possibility than the "strong" hypothesis (w/ Japanese, Korean, etc). This paper [6] has a very cool approach to evaluating the weaker form of the hypothesis that I think most HN readers would find interesting.
So to answer your question, yes, I believe your assessment is correct, and mainstream historical linguistics does as well. But I wouldn't go so far as to say there are any nails in the Altaic hypothesis' coffin just yet.
I guess at this point similarities between the two languages are assumed to be a result of regional influence?