Without the Apple's ad, wouldn't there be some _other_ ad instead? Presumably a competitor's, or the brand's own, paying the Google tax?

Yes, but if their own ad converts the user to a paying customer, they don't pay 30% of their revenue to Apple.

Yes, but an ad to a competing product would become very expensive, as it wouldn’t convert and it’s cost (cpc) would increase as it is deemed less relevant.

Or the the developer would run their own ad.

But those are very different propositions to a middle man running a ‘genuine’ ad which simply takes a cut.

The example shows the brand name being ranked 1st. For this to be a poor result for the brand it would require cannibalisation rates to be high enough to push the 15/30% fee higher than their margin - that’s not realistic because the app store price would never be at a loss to begin with (with many alreadying increasing the app store fee to match the 15/30% cut.) In all likelihood the end result is a net profit for the developer at the expense of their competitors.

Also neglected for consideration: the developer doesn’t pay for these ads - and the scarcity isn’t high enough to meaningfully affect bid pricing over regular competition.

It doesn't need cannibalisation rates to be high enough to push the fee higher than their margin to be worse off, it just needs (delta between direct and App Store margin) * cannibalisation to be greater than the opportunity cost of users who would have given up trying to subscribe to HBO if they hadn't seen Apple's search result [assuming click costs/volume differences roughly even out]. I don't think it's exactly difficult for people to find a link to HBO Max from those results if the one that says "Download the HBO App" isn't there

Sure, they still make profit from the Apple signup, but less than they would have done if people were signing up from their website.

Needless to say if the App Store price is increased 1:1 to match so the developer definitely doesn't lose, it's a terrible deal for the consumer instead (if you're Googling, it's not even like Apple can claim the App Store is adding value with simplicity of discovery...)