I do think that a substantial part of the issue here is terminological, or similar. But not a mysterious mathematical issue (beyond a standard sort of example).

Namely, first, indeed, convergence, or even absolute convergence, of the series $\sum_n \int_n^{n+1} f(x)\;dx$ proves nothing about any sort of convergence or integrability or absolute integrability of $f$. The example of $f(x)=\sin 2\pi x$ illustrates this, and it's not a pathology, etc.

On another hand, equally robustly, if "$f$ integrable"$ means _absolutely_ integrable, then by various means $\int_1^\infty f = \sum_n \int_n^{n+1} f$.

If real-valued $f$ is "integrable" in a technical Lebesgue-theory sense, namely, that one or the other of its positive and negative parts is absolutely integrable, while the other may fail to be so... we still reach the conclusion.

If there are other contexts of interest to the questioner, I'd be happy to entertain them. But, beyond the basic counter-example to a too-naive appraisal of the situation, I do not think there's a pathology here.

I do think that a substantial part of the issue here is terminological, or similar. But not a mysterious mathematical issue (beyond a standard sort of example).

Namely, first, indeed, convergence, or even absolute convergence, of the series $\sum_n \int_n^{n+1} f(x)\;dx$ proves nothing about any sort of convergence or integrability or absolute integrability of $f$. The example of $f(x)=\sin 2\pi x$ illustrates this, and it's not a pathology, etc.

On another hand, equally robustly, if "$f$ integrable" means absolutely integrable, then by various means $\int_1^\infty f = \sum_n \int_n^{n+1} f$.

If real-valued $f$ is "integrable" in a technical Lebesgue-theory sense, namely, that one or the other of its positive and negative parts is absolutely integrable, while the other may fail to be so... we still reach the conclusion.

If there are other contexts of interest to the questioner, I'd be happy to entertain them. But, beyond the basic counter-example to a too-naive appraisal of the situation, I do not think there's a pathology here.