GTM and redirect DNS request (server maintenance page) BASIC QUESTION

There's no need to assign a Last Resort pool and to use the Global Availability method with two pools. It is sufficient to use one or the other. If you are attempting to have a fallback/maintenance pool, I recommend simply configuring the Last Resort pool. Here is a sample configuration snippet (with some parts omitted for clarity):

gtm pool pool_primary {
    fallback-mode none
    members {
        s200:www01 { }
        s201:www01 { }
    }
    monitor gateway_icmp 
}
gtm pool pool_backup {
    fallback-mode none
    members {
        s222:www01 { }
        s223:www01 { }
    }
    monitor gateway_icmp 
}
gtm server s200 {
    virtual-servers {
        www01 {
            destination 10.10.212.200:http
        }
    }
}
gtm server s201 {
    virtual-servers {
        www01 {
            destination 10.10.212.201:http
        }
    }
}
gtm server s222 {
    virtual-servers {
        www01 {
            destination 10.10.212.222:http
        }
    }
}
gtm server s223 {
    virtual-servers {
        www01 {
            destination 10.10.212.223:http
        }
    }
}
gtm wideip www.example.com {
    last-resort-pool pool_backup
    pool-lb-mode round-robin
    pools {
        pool01 { }
    }
}