The short range missiles don't particularly concern the US and don't move the needle in US politics.
That could in fact be a feature, not a bug, particularly when their southern brethren are embroiled in a war of policies with Japan and Japanese. Such missile launches could be interpreted as solidarity between Koreans in various ways.
Its a really complex situation. The ambiguity creates arrows that point in all sorts of directions. Standard Japanese TV media put some emphasize on the counter view that after Moon's sharp rhetoric against Japan such as things like "economic invasion" "will not lose to Japan a second time" afterwards there was his "willing to extend a happy hand". What has also been broadcasted is that there were demonstrations against the Moon administration as well, and showing some interviews with South Koreans on the ground saying things like that they don't participated in the "boycott Japan but feel nervous to by anything Japanese", or "Japan has been a nice country, how could a president say things like that", and so on. Right wing media of course spend more time on the anti-Japanese stuff. Some of the demonstrations against the Moon administration had a number of American flags along with the Korean flags. These likely made with the ROK conservative wing which likely would be in full support of the US-ROK joint-training and are generally more hard line in approach towards DPRK. I could certainly imagine Moon himself not liking the US-ROK joint-training and only allowing it for other pressure reasons such as perhaps political because over the last year or so, he did lose some approval rating points because of his soft approach to DPRK, dipping below 50% at around May. 50% is still relatively good for other leaders in other countries but considering that Moon became president with something like an 85+ approval rating, dipping below 50% was a losing trend. But soft approach to DPRK wasn't the only point that cost him some approval points. The other was jobs. Under his economic programs, senior citizens job rates increased, but the younger graduate age bracket of 18-29 years of age still has a very high unemployment rate. 1,000s of young South Koreans have even gone to Japan looking for work (which is one of several points that I think some Japanese need to be careful about not getting too angry in their response to ROK rhetoric with the history and all since getting some young skilled South Korean workers is a benefit to Japan). So the unemployment with the young age has also resulting in demonstrations again Moon. One other pressure point probably coming in from another angle was Moon's failure to stick with a campaign pledge to raise the minimum wage a certain amount, falling around halfway short if I'm not mistaken. All in the meantime, South Korea's fertility rate has falling below 1.0 it seems, makes Japan's 1.42 or whatever it is look really good.. While Moon's approval rating did pick up a little after his anti-Japan rhetoric following Japan's decision to reduce ROK from the white list of countries with easy free trade, climbing back up a little above 50%. However part of his response to Japan's response was things like removing barriers to overwork hours for research companies with the aim of developing the tech to reduce dependency on Japanese tech. This whole mess they put themselves into is deriving from the liberal ROK wing's recent idea that the 1965 money aid from Japan for the forced labour is no longer sufficient. They also doubled down on the comfort women stuff again recently. But this noisy side makes up around half the population from my observation. A quarter is general neutral don't care and play along with the majority. The last quarter actually has no problem with Japan but keep their thoughts on the whole matter in the closet to keep themselves safe from the mob. Things are not so simple as you describe and knowing about these specifics makes a difference in having a better approach.