I guess at this point similarities between the two languages are assumed to be a result of regional influence?
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.
[0] https://academic.oup.com/jole/article/3/2/145/5067185
[1] https://www.academia.edu/4208284/WHY_JAPONIC_IS_NOT_DEMONSTR...
[2] https://www.academia.edu/6345901/The_end_of_the_Altaic_contr...
[3] "non-traditional" = anything outside the comparative method (https://en.wikipedia.org/wiki/Comparative_linguistics). See eg: https://www.reddit.com/r/linguistics/comments/6tg6cr/why_is_...
[4] https://pure.mpg.de/pubman/item/item_2564924_3/component/fil...
[5] https://www.cambridge.org/core/journals/evolutionary-human-s...
And, as the post points out, the whole line of argument is a curiosity that has little concrete bearing on the fact that the two are very similar from a learner's perspective.
The two languages are very divergent, and the usual mechanisms for classifying languages are difficult to apply because the dominant Chinese influence came to each separately after they diverged.
It's even more complicated than that. In addition to the Ainu there were the Jomon people, a non-agricultural but nonetheless settled society who are believed to be the first people in the world to use pottery - which is surprising for a non-agricultural people.
They might in turn have been an offshoot of the first migration of modern humans out of Africa along the south coast of Asia, whose haplotypes [1] occur today at some of the highest frequences in Japan and as far north as Mongolia and as far south as Australia. Some sub-branches spread west all the way to the edge of Europe.
This pattern - early hunter gatherer populations forming the substratum that later mixed with larger migrations of agricultural populations - is not unique to Japan.
It's quite similar to what you find in the paleogenetics and history of most regions of the world, including Europe and South Asia.
We're back to population migration / replacement theories being ascendant rather than the situation 30 years ago when cultural diffusion theories were preferred.
From https://www.nhm.ac.uk/our-science/our-work/origins-evolution...
"When we look at genetic variation in modern British people today, we find that – for those who do not have a recent history of migration – around 10% of their ancestry can be attributed to the ancient European population to which Cheddar Man belonged. This group is referred to as the western European Mesolithic hunter-gatherers. However, this ancestry does not relate specifically to Cheddar Man or the Mesolithic population of Britain. Well after Cheddar Man’s death, two large-scale prehistoric migrations into Britain produced significant population turnovers13. Both of these migrations into Britain represented westward extensions of population movements across Europe10-12. In both cases, these migrating populations intermixed with local people who carried western European Mesolithic hunter-gatherer ancestry, as they moved across Europe. When these populations arrived in Britain they already had some hunter-gatherer ancestry derived from this mixing with local populations. Therefore the majority of western European Mesolithic hunter-gatherers ancestry that we see in modern British people probably originates from populations who lived all over Europe during the Mesolithic, which was carried into Britain by these later migrations."