To add further context--I'd like to emphasize that an understanding of written Chinese would help with Kanji, but like you said, to a limited extent. When reading Kanji, there are cases where you'd have to be cognizant of Onyomi and Kunyomi (Basically pronunciations rooted in Chinese vs. Japanese). Not as important if you are strictly "reading", I suppose. However, this would also not provide insight when reading Hiragana nor Katakana, how particles are used, rules for conjugation (polite vs. casual, past vs. non-past tense, etc.), further reducing mutual intelligibility. In some cases, Chinese characters may be visually identical to Japanese kanji, yet have different meanings or applications. Traditional Chinese vs. Simplified Chinese is also a whole other topic.

Examples where there is some similarity:
JP: 走る
EN: Run (verb)

CN: 走路
EN: Walk (verb)

Matching characters, unrelated meaning and application:
JP: 勉強
EN: Study (noun)

CN: 勉強
EN: Reluctantly (adverb)

Furthermore, Chinese uses Subject-Verb-Object (SVO) word order, whereas Japanese uses Subject-Object-Verb (SOV) word order. Japanese also regularly uses subject omission, so it's important to consider these things if you're moving from one language to the other. Missing an understanding of these differences could lead to pretty different interpretations of a sentence.

That being said, having a background in Chinese would be more beneficial when picking up Japanese than the other way around, IMO.