They gain the goodwill of a prospective employee who might cycle back and be a great fit later on in their career (and who has more skills then). It's also about what they don't lose, which is not only me but potentially other prospective candidates. Know what happens when someone mentions they are considering applying to a place that never got back to me? I tell them don't apply there, they waste your time / they don't care about candidates / their interview process sucks. How many people does that person then go tell? I've absolutely punted on applying to places which my friends have said similar things about.
And providing feedback is very likely to hurt your reputation. One major company did give me feedback, and I've made fun of them endlessly for it - they explained that the manager loved me, I did great on systems design and architecture, but I couldn't pass the hiring committee because I didn't do well enough on tree traversal tricks.
Even that in a way is useful to them. If they are constantly getting confused, bemused, or angry responses to what they think is honest feedback, it is a red flag about their hiring process.
Every hiring process under the sun has a large, vocal group of people who think it's shit-tier. (Some people like me are equal opportunity haters - I've never seen a process that didn't have stupid flaws.) If you give feedback, eventually the group for your hiring process is going to find you.
Reputation only matters if your candidate pipeline isn't full enough. For example, Google is notorious for their awful hiring gauntlet, but people still line up around the block to work for them. If your pipeline is fine, reputation and goodwill from people you rejected are worthless.
Most companies are not Google, so their (local) reputation matters a lot.
I'm a .Net developer. In my area there is a well known company that hires a lot of .Net devs. I heard from 2 people that .Net devs are basically treated as 2nd class citizens (after Delphi devs). So now I will not consider applying there. I don't know if they would want to hire me, but I have nothing to gain finding out.
Reputation matters a lot.