I don't know if many topics have been both overthought and underthought to the degree this one has.
If you begin with this intro to LuckyGunner's ballistic tests,they will give you an address to the web page where they show test results for 100+ loads in 9mm,40 S+W,and 45 ACP.
If you will do this research,you will gain a much better answer to your question than can be described on a forum page
https://youtu.be/0ZIYMTlf19A
I don't think it is accurate to say "there is no difference".I can throw rocks in a pond and see the difference.I think some of the high speed video reveals some rounds have more bounce and "wave action" in a gelblock than others.
I think that is "temporary cavity effect".Generally,at handgun velocities,temporary cavity does not produce much,if any,additional trauma,blood loss,etc.
Between the trauma surgeon,coroner,and measureable effects in gelblocks,I believe the charts,graphs and data will show similar results 9mm,40,and 45.
And most of the expanding bullets work well.
I think they will all kill about the same,subject to infinite variables.
My opinion believes with comparable bullet performance ,a 950 to 1100 fps 185 gr or more expanding 45 ACP bullet will provide more temporary distraction to a bad guy than a 115 or 124 gr 9mm bullet(CNS exception).
"Temporary" is hard to measure,and live target reaction is subjective and inconsistent.
Its hard to quantify as data,for data driven decision making.
But watching the gel blocks on high speed...
IMO,the temporary results will get a temporary effect on the bad guy
I would think he would feel more "punched" ...momentarily. Some of the time,with the more major cartridge/load.
It might only last a few seconds,but those might be important seconds.They are seconds the bad guy cannot shoot well,and they are seconds of blood loss.
I'm saying "Temporary cavity" may be "temporarily disabling",yet not show as permanent wound channel.
The tradeoff,between controllability and mag capacity,might be 2 hits of .45 vs 3 hits 9mm.And that makes a good argument.