Should Cache-Control request headers influence server-side caching?

A have seen a lot of discussion regarding proxies with regard to RFC7234 and the Cache-Control header, but I'm not too clear on whether the request headers should impact server-side caching or not.

The diagram below shows where the headers I am concerned are located (down arrows are request headers, up arrows are response headers):

      ┌────────┐
      │ Client │
      └────────┘
        ↓    ↑
      ┌────────┐
      │ Cache  │
      └────────┘
        ↓    ↑
      ┌────────┐
      │ Cache  │
      └────────┘
        ↓    ↑
      ┌────────┐
      │ Server │
      └────────┘
  ??? → ↓    ↑
    ┌─────────────┐
    │ Server-side │
    │    Cache    │
    └─────────────┘

I have a number of questions regarding the interaction between Cache-Control and server-side caching (e.g. should Cache-Control: no-cache, mean that the server should re-validate its own cache?). But the vast majority boil down to the following:

Should the server-side use Cache-Control and other associated headers from the request for logic regarding its server-side caches?

Take the following small (albeit crude) Python Flask application as an example:

@app.route('/app/<id>')
def endpoint(id):
    if 'no-store' not in request.headers['Cache-Control']:
        # If 'no-store' isn't specified, try cache first
        entry = entry_cache.get(id)
        if entry is None:
            entry = service.get_entry(id)
    else:
        # Otherwise, immediately talk to the service
        entry = service.get_entry(id)
    return entry

In addition, this raises another question which, depending on the answer, may render the above irrelevant:

What kind of caching should be done on the server-side? What situations should we rely on HTTP caching and calculate the response on each request (outside of situations like If-None-Match)?

For context, most applications I tend to develop are designed as micro-services either as containers or as serverless functions with caches tending to be calls to external services such as Redis. Not many browsers call these HTTP endpoints; most clients are business applications.

Update (2021-08-17): Thanks Kevin for the feedback! I have updated the diagram and some of the wording to clarify that I refer specifically to the request caching headers rather than the response headers. In addition, I've added a very small (and very crude) Python Flask app to help illustrate the type of logic I am referring to.