Integration to Salesforce through REST API without using consumer secret/key

Definitely doable. The admin should not have to customize anything:

1. create your Connected App, once, in an org with a managed package and namespace,
2. bake that exact client_id into the runtime code that you distribute,
3. use Web Server authentication flow or Username-Password flow when calling REST API,

Maybe there is a misunderstanding of how Connected App really operates here. Imagine a Salesforce Package as an analogy. You control the true source code in a Packaging / Release Org but you deploy or install a special instance of it. Similarly with a Connected App:

• you control the Connected App 'definition' including client_id and secret,
• a user allows (or a sysadmin can manage) the connected app 'instance' when it's used.

Does this help?

It's pretty similar with Salesforce, but your sample code has the parameters incorrect for the platform. See OAuth 2.0 JWT Bearer Token Flow from the Salesforce documentation for more details. Rather than using the JWT for authorization on REST requests, you'll exchange it for an access token, which you'll then include with a Bearer authorization header (if you're constructing your REST requests manually).

I use JWT in Python in one of my projects and have a Gist available showing how to use it with simple_salesforce. It's short, so I'll include the entirety here:

import jwt
import requests
import datetime
from simple_salesforce import Salesforce
from simple_salesforce.exceptions import SalesforceAuthenticationFailed

def jwt_login(consumer_id, username, private_key, sandbox=False):
    endpoint = 'https://test.salesforce.com' if sandbox is True else 'https://login.salesforce.com'
    jwt_payload = jwt.encode(
        { 
            'exp': datetime.datetime.utcnow() + datetime.timedelta(seconds=30),
            'iss': consumer_id,
            'aud': endpoint,
            'sub': username
        },
        private_key,
        algorithm='RS256'
    )

    result = requests.post(
        endpoint + '/services/oauth2/token',
        data={
            'grant_type': 'urn:ietf:params:oauth:grant-type:jwt-bearer',
            'assertion': jwt_payload
        }
    )
    body = result.json()

    if result.status_code != 200:
        raise SalesforceAuthenticationFailed(body['error'], body['error_description'])
    
    return Salesforce(instance_url=body['instance_url'], session_id=body['access_token'])

Call jwt_login() with the consumer Id, username, JWT signing key, and a Boolean for sandbox or production and you'll get back a simple_salesforce.Salesforce instance. If you aren't using that library and just want the access token, you could return body['access_token'] instead and modify the error handling.

N.b. as written this requires Python 3.6 for the time handling.