That’s a great observation!
Yes, 品味 and 咀嚼 have quite similar meanings when it comes to thinking thoroughly about things, just like tasting food carefully. But they’re still somehow different. Come back to the food analogy, we often “品味” tea, wine, but not food that requires chewing; similarly, we 咀嚼 food but not drinks. So 品味 is more like to appreciate, whereas 咀嚼 means that it still requires another procedure–to chew. When one 咀嚼 something, one can be critical, but when one 品味 something, one simply enjoys and appreciate it. We can make two different sentences:
Indeed, I believe Chinese people do associate eating with thinking. In addition to the above two examples, the idiom 囫囵吞枣 literally means to swallow, but it also applies in situations where people accept things with thinking thoroughly.
There is another saying：猪八戒吃人参果，全不知滋味。It means swallowing precious food (人参果）without knowing its taste (不知滋味). This also describes people who want too much but cannot get anything out of it.
I hope this helps! Thanks for the thoughtful question.