So I've been churning all this over to try to make a story as I like telling stories and I think there is one to tell here.
Let's start with 1 given point; that the White Sox's player evaluations are totally, completely, broken. Let's go from there.
The White Sox put together their budgets and put Machado at $175 and Harper at $225. They realize they can afford both of them at those values. In December they meet with Boras and put an offer similar to that on the table, they meet with Machado in late December and put that offer on the table. Boras doesn't give much of a response but they hear out the White Sox's presentation. Machado's side starts negotiating. They bring in the "Much more experienced" negotiator at some point, and sometime in January they start adding vesting options.
By mid-January the White Sox have a 7/$175 offer on the table to Machado and they've added 3 vesting options that could bring the value to $250, but they don't understand how those vesting options are viewed or how low that dollar amount is. They think that is a huge dollar amount, so they leak their offer to Bruce Levine thinking that it's so big it will scare away the competition, and Bruce doesn't get all the details but he also hears that the amount could go up to $250, so he's on the radio giving that classic interview about how the contract could be worth more.
The White Sox think the $175 million offer will scare teams away, instead suddenly teams realize that this isn't going to go well over $300 million and they start sniffing around. Other offers start appearing, so Lozano's side goes to the White Sox and says they're going to have to guarantee that money just to get into the conversation. The White Sox gradually increase their guarantee, including right before Soxfest where they think they've done what Lozano asked and they think they're going to get it wrapped up at $200 million or so. Lozano still understands that's no where close to the final total, but every time the White Sox up their offer they think they're doing Machado a favor and bidding against themselves.
Meanwhile, the White Sox keep leaking that they think they have a shot at both of them because they could do both for $225 million deals, but Scott Boras has stopped taking their calls because they're so far out of the race they're not worth hearing from, so they haven't talked in weeks.
Over the last few weeks the Padres get more serious, realize that this isn't going to break their bank, their GM works with ownership to get ownership on board, and they come in with a strong $280 million guarantee last week.
The White Sox up their bid to $250 million guaranteed. They think "They've done everything Lozano asked them to do" because that was something he asked them to do in January - Rick Hahn's words. When the $280 million offer comes through, the White Sox go ahead and add in 2 vesting options that they are convinced are valuable to the player, failing to understand how weak those are from the player's side.
When Lozano comes around and says "I'll sign at $300", the White Sox don't even think it's worth talking about because they're convinced they've been bidding against themselves. That pushes the Phillies out, the Padres say yes, and he's a Padre.
Meanwhile, everything the insiders heard was true - that the White Sox had a budget for Harper and Machado, that Machado's budget was less, and that they didn't want Machado's money to get into the Harper range, but they thought they were doing Machado a favor by going into that range. Meanwhile, they don't realize nearly how far away they are, and when other teams realize they're scooping up good deals, the White Sox go into "We've got to pay for the rest of our roster" excuse mode. So, we still hear some of the "They have a budget for Harper that was higher than Machado" stuff, but that's because they never realized that the budgets they had for both of them were laughably low. They thought that a $175 million offer would scare teams away, but it did the exact opposite, and they thought they were bidding against themselves as they went through their own ceiling to put that $250 million offer on the table for Machado.
I don't know if it's right, but that's basically every single leak this offseason fit into one tidy picture.