How is Madueke position right now at Chelsea any different, he's no longer occupying his favorite right flank spot which is now fully Pedro Neto's operating area, but if anything in Arsenal he's just got to occupying the left flank like he currently does at Chelsea, possibly ahead of Martinelli. And more so Saka have been struggling to stay injury free lately, so perhaps, playing time is surely not going to be his problem at Arsenal, and I don't think Arsenal are going to spend well over 50 million pounds on him with the intention of playing backup, except he doesn't live up to their expectation.

Seeing that Berta has seriously talked to multiple other targets ^{[e.g., Cristhian Mosquera and Eberechi Eze]} makes me believe Arsenal isn't on a tight budget and IIRC, they didn't sign anyone in the previous January transfer window so there's plenty of money, but for some reason, they still want to save that £5M ^{[if I were them, I'd pay it right away to sign Gyökeres and then recoup portion of it through shirt sales]}.

Ok let's say this is the reason why they got Madueke but what happened to Rodrygo? He's better than Madueke by far or what of Eze? Arsenal have been running to get for some time now or are they not see Adingra who just signed for Sunderland? Like the transfer market have better players that are more productive than Madueke and no one will tell me that is because the price was cheap that's why.
If they are waiting for Sporting CP to accept what they are offering then they should keep waiting till the player Gyökeres decide to change his mind to play for another club not the Gunners. To be honest this Madueke move is never going to help the Gunners in any way unless Arteta will convert him to a striker...who knows very soon Jackson will be moving to join Madueke.