Mine sometimes does. We also tend to refund recent payments liberally if, for example, someone who hasn't been using the app in a while asks to cancel just after they've renewed for another period.
Sure, it's a principle thing, treat our customers as we'd like to be treated ourselves. But it makes sense from a purely business point of view as well.
In return for giving up some small amount of subscription revenue by putting through a cancellation that was going to happen anyway a bit sooner, we generate positive sentiment. We often get a nice thank-you message back, with extra information about why someone was cancelling or hadn't been using the app recently.
We know for a fact that we have sometimes gained new customers from referrals by those people we helped out a little. Sometimes a former customer's situation changes again later and they come back to us, too.
And the reality is that particularly with online payment methods, there's also a small risk that someone will do a hostile chargeback without bothering to even try cancelling, particularly if they forgot about a subscription and changed their email address or something like that. Even if you've done nothing wrong and provide evidence accordingly, you have a good chance of losing a dispute anyway, and one way or another it ends up costing you more than it would have done to preemptively cancel a subscription that you knew wasn't being used for a long time.
This is from the perspective of a small business in a niche market, where there is definitely a sense of community and reputation does matter. We've met some of our customers in person, and there are many more mutual connections or friend-of-friend kinds of relationships that might be relevant one day. Maybe things work differently when you're running a huge business with a strong brand in a huge market; I've never done that, so I wouldn't know. But I honestly can't imagine why we'd want to run things any other way. There are few things more valuable to a business like ours than a good reputation in the community we cater for.