Jointly training a speech enhancement (SE) front-end and an automatic speech recognition (ASR) back-end has been investigated as a way to mitigate the influence of \emph{processing distortion} generated by single-channel SE on ASR. In this paper, we investigate the effect of such joint training on the signal-level characteristics of the enhanced signals from the viewpoint of the decomposed noise and artifact errors. The experimental analyses provide two novel findings: 1) ASR-level training of the SE front-end reduces the artifact errors while increasing the noise errors, and 2) simply interpolating the enhanced and observed signals, which achieves a similar effect of reducing artifacts and increasing noise, improves ASR performance without jointly modifying the SE and ASR modules, even for a strong ASR back-end using a WavLM feature extractor. Our findings provide a better understanding of the effect of joint training and a novel insight for designing an ASR agnostic SE front-end.