I've done both many times. Almost all bin foundations we pour on the farm are monolithic slabs with the footer and slab all in one pour. Most buildings we do separately but mostly because there are complexities that make it easier. Sometimes getting the footings deep enough here without the sides of the trench collapsing, messing with under slab insulation and tying everything together is just easier on larger buildings if broken out separately. As I recall, this is simply a small replacement addition and probably has nothing coming up from underneath so I don't see why it can't be done as a monolithic slab.
I don't really see an advantage or disadvantage one way or another other than ease of achieving your overall objective which if the person doing the concrete is going to cut 3K off your bill, I would definitely go that way. Cracking shouldn't be a concern. Either way can crack depending on soil conditions, quality of the cement pour (drier it is poured the better and the slower it cures the better), etc. Almost all foundations crack because they don't have regular relief joints cut into them like say a sidewalk or a driveway. The key is that they have proper reinforcement in them to prevent them from moving in relation to each other. Every room in my full size basement (under a ranch house) has micro cracking. It happens as the concrete cures and shrinks. As long as there is a proper bed underneath the concrete, this shouldn't affect anything especially since if I recall, this will act as a crawlspace floor.
If you ever have to tear it out for some odd reason in the future, you aren't going to save much money if at all by pouring the footer separately. The work involved in being careful to not disturb the footer while tearing out massive chunks of concrete would slow down a project such that most would probably just want to tear it all out and repour it.
Thinking about this a bit before posting, I should add that I if you do this in two pours, i.e. footer and slab separately, I would end the footers at the bottom of the slab level and pour the slab over the footers so that you have a nice flat surface across the entire slab. Sometimes, people pour footers high up to be part of the wall and then fill in the slab later. I see this a lot in basements and garages, mine included. Going this route is fine but I recommend isolating the slab from the wall/footer with some isolation barrier. This prevents say a settling wall/footer from hooking the slab (if the concrete was poured directly up against the wall/footer) and jacking it around. Whomever poured my garage floor of this house made that mistake and one corner of my garage was built on fill and settled about three inches. It caught the concrete and was essentially using my garage slab as a a giant lever until it broke. From high to low point in my garage when I moved in was about 12 inches of elevation. I ended up jacking up the corner of my garage to adjust for the settling since it was 40 years old at that point and not likely to settle any further. I repoured the garage slab using isolation material and haven't had any problems since.
