tl;dr: You may want to send an email to the editor about this. Although it will not change anything about this paper, it will be a good lesson for the editor to engage in more constructive correspondence with referees who are spending a lot of effort for free to enable the existence of the journal itself, as well as to be more responsible in his/her decisions next time.
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This is something that can and does happen in reputable and even prestigious journals. We do not have the full picture, only what you wrote, but based on my experience, I tend to believe you are correct, and the editor made a judgment that takes into account non-scientific factors such as:
Efficiency. He/She does not have time to deal with the details too much. He/she needs to make a fast decision and it's safest to let the paper in because it was on a fast pace track anyway.
Politics/Importance/Perceived-importance of papers/author. Since it was on a fast track the paper was probably important for some reason to the journal/editorial board. Maybe it gives them some prestigiousprestige? Or whatever reason. The editor knew there is a reason for concern, but went with publishing it because he/she decided to ignore what they perceive as "details" that "do not take into account the whole picture", or something like that.
Possibly, the editor did a genuine decision, believing the paper merits acceptance, and that "you are just picking on the details". They may have a different view than yours, they may think that details are unimportant.
Overall, I tend to agree with your view: details are extremely important and decisions should be made based on objective merits solely as much as possible. Unfortunately, that is not how the system works.
Conclusions: You may want to fight a bit over it. I don't see it as harmful. Simply send an email to the editor to inform them that you think they made a wrong decision as long as the reviewers don't address your concerns. This will not change anything for the present paper, but for the next paper this editor will be more cautious I assume. He/she may be a bit pissed off by your email, but so be it.