I don't completely agree with you because the performance of Lukaku is this season is not good enough to say he worth trying hard and paying a huge amount of money for him, Lukaku was injured in many games and he couldn't help his team for weeks, however after getting recovered Lukaku could show in the last week but all we saw from Lukaku was just in a few last weeks which is not enough to say he had a good performance.
After all this, if the team still finds him worthy of signing then they can see some potentials in him that’ll help their team grow to new heights. If the amount Chelsea is asking for is very big and the player is not worth it, they could have backed out of the deal but they keep pushing and hope that Chelsea will reduce the said amount to a lesser one. This is football and there’s business in football. In as much as you want a player very desperately, you’ll be ready to pay extra fees than what he’s worth in order to have him by your side, I think that’s just the case here between Inter Milan and Chelsea about Lukaku.
I don't think there is anything for inter milan to fight for after they threw in the towel or withdrew for Lukaku. A strong reason for me why Inter Milan is doing this is because they have found the right replacement for Lukaku for next season. Marcus Thuram is much better than Lukaku if we compare their stats last season so that's what made Inter milan back off to get Lukaku.
So on the Ac milan side, they have now succeeded in recruiting Tijjani Reijnders from Az where he will soon complete his medical and join ac milan this season. I see Ac milan being quite active in this transfer window, Ac milan previously got Loftus-Cheek, Pulisic and several other players who were recruited on free transfers. Pioli really wants to rebuild AC Milan and I think he's got some of the players he wants.