I've had some bad experiences with recruiters, both in-house and otherwise, and I feel they usually get screenings wrong. But there's an even more important reason for you to take over this process now: you don't know where your dropoff is.
You need to get familiar with what's going on. Are you not getting any good resumes? If so, then the problem is your outreach. Are you getting lots of great resumes but they drop off after the first call? Then the problem is your sales pitch on the call.
If it's your responsibility to fill this position, then it's time to get your hands dirty in the data and figure out where the problem is.
Without more data on where you have dropoff, these are my takes on your process:
- The other commenters in this thread are right: the 4-hour assignment is likely a huge issue. I'd be shocked if you didn't have a huge dropoff there. There's no incentive for senior engineers to do this type of thing, because so many companies want to hire them. It's better to take your chances on someone who seems good, and be prepared to fire them if it turns out you're wrong.
- You should put a salary range in the job posting. You said "benefits and salary are good", but literally the only reason to not list these things in a posting is if they're not actually that good. You may need to pay more for someone very senior. They may be expecting more -- even substantially more -- than the person who just retired.