Apigee HTTPBasicAuth timing out

So I have this working request that uses a HTTPBsaicAuth that needs to go through Apigee. I configured my Apigee proxy with a Basic Auth to retrieved the username and password in the headers However I continue to get the error:

{"fault":{"faultstring":"Gateway Timeout","detail":{"errorcode":"messaging.adaptors.http.flow.GatewayTimeout"}}}

Here is the original request without the Apigee pass through that works:

import requests
from requests.auth import HTTPBasicAuth
url = "https://myapi.com/path"
auth =  HTTPBasicAuth("username", "password")
response = requests.get(url, auth=auth, verify=False)

The request above works without issues. My new request, including the Apigee url and apikey within the headers:

import requests
from requests.auth import HTTPBasicAuth
url = "https://apigee-proxy.com/basepath/path"
headers = {
    "apikey" : "XXXXXX",
    "username": "username",
    "password": "password"

}
response = requests.get(url, headers=headers, verify=False)

Proxy setup, very basic:

<ProxyEndpoint name="default">
    <Description/>
    <FaultRules/>
    <PreFlow name="PreFlow">
        <Request/>
           <Step>
              <Name>Extract-Variables-1</Name>
           </Step>
           <Step>
              <Name>Basic-Authentication-1</Name>
           </Step>
        <Response/>
    </PreFlow>
    <PostFlow name="PostFlow">
        <Request/>
        <Response/>
    </PostFlow>
    <Flows>
    </Flows>
    <HTTPProxyConnection>
        <BasePath>/basepath</BasePath>
        <Properties/>
        <VirtualHost>default</VirtualHost>
    </HTTPProxyConnection>
    <RouteRule name="default">
        <TargetEndpoint>default</TargetEndpoint>
    </RouteRule>
</ProxyEndpoint>

UPDATED: Extract-Variables-1.xml:

<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
<ExtractVariables async="false" continueOnError="false" enabled="true" name="Extract-Variables-1">
    <Source>request</Source>
    <Header name="username">
        <Pattern ignoreCase="true">{username}</Pattern>
    </Header>
    <Header name="password">
        <Pattern ignoreCase="true">{password}</Pattern>
    </Header>
    <VariablePrefix>auth</VariablePrefix>
    <IgnoreUnresolvedVariables>false</IgnoreUnresolvedVariables>
</ExtractVariables>

Basic-Authentication-1 policy:

<BasicAuthentication async="false" continueOnError="true" enabled="true" name="Basic-Authentication-1">
    <DisplayName>Basic Authentication-1</DisplayName>
    <Operation>Encode</Operation>
    <IgnoreUnresolvedVariables>false</IgnoreUnresolvedVariables>
    <User ref="auth.username"/>
    <Password ref="auth.password"/>
    <AssignTo createNew="false">request.header.Authorization</AssignTo>
</BasicAuthentication>

Default route rule:

<TargetEndpoint name="default">
    <Description/>
    <FaultRules/>
    <PreFlow name="PreFlow">
        <Request/>
        <Response/>
    </PreFlow>
    <PostFlow name="PostFlow">
        <Request/>
        <Response/>
    </PostFlow>
    <Flows/>
    <HTTPTargetConnection>
        <LoadBalancer>
            <Server name="my_target"/>
        </LoadBalancer>
        <Path>/</Path>
    </HTTPTargetConnection>
</TargetEndpoint>

The target server environment variable "my_target" = myapi.com, port=443

Is there something I am missing? Any help is appreciated.

UPDATE: Proxy now includes Extract-Variables-1, I can see the Authentication header being applied, but still timing out..:

UPDATE: The solution @Eddoasso offered worked for the BasicAuthentication. The connectivity issue has to deal with a firewall within my organization, thank you all for your help.