Abuse@namecheap.com is indeed the best channel. Our Twitter reps are quite responsive too. The abuse team reviews every case. In some cases, it can also be effective to reach out to the "host" too.
Does the abuse team always reply to abuse reports? I've had one open since the 5th with no activity and I'm unsure if it is just being ignored.
I have reported a large amount of phishing domains and never received a reply, they also never got taken down.
There is a backlog. Like I said, they're a very busy team and are quite thorough in their review. Feel free to ping @namecheap on Twitter with the ticket number and they may be able to help escalate, depending on the severity of the case.