Even with the immediate passing of control, the premise still works, because women perform the first pass of filtering, giving them an opportunity to preemptively reject the most undesirable of the pool before they have a chance to reach out. It's also an effective tool to fight the sheer volume of spam.
The same kind of invitation plays out in real life with eye contact and body language.
Tinder and Bumble (and Coffee Meets Bagel, the League, etc.) have each party approve the other. The only major thing that distinguishes Bumble from Tinder is the female-message first policy, which is often side-stepped as described above.
I can see why you would think that. Based on my experiences actually using both Tinder and Bumble, it's possible that your experiences might differ significantly from mine.
It's not side-stepping. On Tinder, it's very common for someone to match, and then just never say anything or respond. Women's profiles complain about this a lot.
Well, I'd say it's 90% side-stepping. Composing a message takes more work than swiping, especially more customized messages. Just saying "Hi" is not much more work than swiping, and so mostly defeats the purpose.
Maybe saying "hi" isn't much more work than swiping in your opinion, but in reality, it seems that it is because so many people don't bother to do it. Maybe they're waiting for the other person to make the first move, or maybe they just want to collect matches, or maybe they're busy chatting up someone else and don't want to start up something new just yet but still want to keep that person around as a possibility; whatever the reason, I've seen a fair number of profiles where the person complains about matches not saying anything at all.
And I forgot to mention, I've also had the experience many, many times where I've said "hi" myself (actually, I say something a little longer than that for my first message, I try to say or ask something about something I saw in their profile or photos), and then just never get a response, but they never unmatch me either.
The same kind of invitation plays out in real life with eye contact and body language.