If a player is readily available I'd expect them to sign very late on in the window. That way the selling club can play all parties against each other to get the most wedge and the agent (s) can do exactly the same
I think quick(ish) deals are only done now if ludicrous bids are lodged for players who either haven't asked for a transfer a la Bolasie/ anyone Liverpool bought from Southampton or if the player has their heart set on a club
Furthermore I'd suggest that contracts and transfers are so convoluted with image rights, sponsorship, various ownerships and greedy agents (stuff which has totally ruined the sport) that a quick thrashed out deal pre transfer window opening is nigh on impossible
I look forward to being shot down on this one.