This is something more people should do! I still remember the companies where the human touch at the hand off stage was present and have nothing but good will towards such companies. If I wasn't working for a company in direct competition for resources, I would probably be referring people to such companies :) On the flip side, I still remember interviewing with one startup where I thought everything went great, had a really amazing vibe with the hiring team and then suddenly got an automated response the next day with a reject. Would I interview again with that team? Probably not. After all, the bs I got fed about how culture was important to them, It feels rather disappointing that that was all a facade in some way.
On the flip side, of most of the people that I interview, I am not confident that I can honestly say that more than a tiny percentage of those people are objectively bad, it is more that they are not a right fit for my team because the team is looking for people with specific skills and there is a mismatch. This means that a little work on the part of the Engineer could better redirect such people (at least better than a "tech recruiter") to teams for whom such candidates would be a great fit. The nature of the recruiting business is that you will always have false positives. More importantly, people are moving targets: A person who is not a right fit today for your team might in a few years be the right fit for the next team that you have.