The reason why Atlanta have allowed him to go on this coaching search could be down to them trying to get something in return for a head coach that they might not end up keeping long term. If you can get a second round pick for him, or even cash considerations, that is better than having to dig out the checkbook and fire Budenholzer sometime during the remaining two seasons on his contract. Trading him costs you nothing, firing him costs you money.
Last year Bud "gave up" the president of basketball operations part of his job. Generally speaking when a person gives up a job like that it is usually the case they have been given a choice between stepping down on their own terms or being fired. Also, Wes Wilcox stepped down from his position as General Manager and became a "special adviser" to the ownership group. Again that screams of step down on your own terms or be fired.
The Hawks decided to make wholesale changes last year when their window closed. From their 60-win team they lost Demarre Carroll (2015), Al Horford (2016), and Paul Millsap (2017) to free agency and traded away Jeff Teague and Kyle Korver in the 2016-17 season. Essentially the Hawks made the right decision to start a rebuild when Paul Millsap decided to depart. Had they kept the team together I think the changes in the front office probably wouldn't have been made. But because those departures did happen and their window ultimately closed, it is possible that their new general manager, Travis Schlenk, who they hired from Golden State could be willing to move on from Bud because he wants to hire his own guy.
As you say in your post, they have four picks in the coming draft. I would go further and say that they do have some talented young players already on their roster. Dennis Schroder just had a career year (19.4 PPG 6.2 APG). John Collins made a positive impression as a rookie power forward and Taurean Prince made strides as he stepped into their starting small forward position (he hit .385% of his three point shots). It depends on how the draft falls but if they came away with Luka Doncic (4th) and say Mitchell Robinson (19th) that could complete their young line up. Alternatively if they draft DeAndre Ayton, Jaren Jackson or Mohamed Bamba with their first pick, they could always retain Kent Bazemore in their starting line up at shooting guard, or look at one of the guard prospects with their second pick in the first round (i.e. Lonnie Walker, Khyri Thomas, Zhaire Smith, Gary Trent Jr etc). This is also ignoring the fact they will have cap space for 2018 free agency and a ton of space in 2019 because then they will only have Bazemore (if he opts in, if he opts out that's an extra $19 million), Schroder, Plumlee, Prince, Collins, Bembry and their drafted players on the books.
So in a sense it might appear strange that they are willing to move on from Bud considering this could be a quick rebuild, but it is not really strange considering he's already been demoted and has to work with a new front office. It is quite common for new front offices in major sports to move on from existing head coaches. So I suspect Bud wants out before they fire him and the Hawks want to move him before they fire him.
