I honestly think your suggestion in the question is probably the best route.
There are major pros for the players if they pick Bob up as an Aspect
If they have Bob as an aspect, it becomes a good way for them to gain fate points by using that aspect to stumble into danger, as well as giving you a pretty good way to introduce plot hooks. If they want to use Bob to augment their die rolls, they could spend a fate point to gain a bonus to whatever roll/reroll (I'd guess lore or something else applicable that's an applicable use of an aspect, but I can't quite recall). Once you've got them hooked on their new Aspect, take it away, and toss them a plot point so they can go get it back.
If they don't grab him as an Aspect, well, you've got a plot device, but they won't be getting any fate from when he decides to throw them a bone.
I would switch between scenarios. (every significant milestone, or around three or four sessions, but mainly whenever the current "dungeon", "book", or "adventure" is done.) I'd suggest taking a look at the books in the series- there are one or two long term issues throughout the books, but by and large each one is a mystery within the book.
What you need to make sure of is that you finish each adventure before handing things over. The villain might not be dead, but he's beaten for now. The city might not be safe, but this particular threat won't be what kills it. (Note: If the bad guys won, which they do sometimes, then that's an ending too.) In the section on cities, it points out that city aspects are expected to change as the game progresses. Basically, whenever one of those aspects changes, and everyone involved in the hubbub has conceded that there's not much more they want to try, then you can safely change. There are a few possibly hiccups.
First, what was the odd character out doing? In Dresden, you all didn't just meet at a bar, so when bad stuff went down and you all pitched together, why did one of you not show up? (That would be the DMs character.) This isn't a hard problem, but you do need to think about it. Did the bad guys start out by neutralizing them? Are they busy dealing with another part of the problem? Are they recovering from a consequence from last scenario? Or, are they right there besides everyone else? (Some people can play a character and DM the game at the same time. It doesn't work very well for me, but if you can pull it off go for it.)
Second, are there any secrets? Is one the the players related to one of the bad guys? Is the big bad working to find the three pieces of the triforce, one each scenario? Are the bad guys secretly breaking the seventh law? Is one of the players secretly a red court infected? (That one actually came up- A player got infected during a session, and hid it effectively for two scenarios. Then we switched DMs when the DM went for a semester abroad, and I had to deal with this sudden new plot thread, and pretending I didn't know when the DM got back. If you're rotating DMs, everyone just has to be okay with the fact that inside of three or four rotations, everyone is going to know whatever deep dark secret they were hiding.)
The best way to fix this is to just not have secrets, but that takes some of the fun out of it. Depending on the players, they could roleplay not knowing. (I love doing that sometimes- the example in the book goes something like "A ghoul snuck in the back window. You failed the alertness roll, so why didn't you notice?") Thing is, just because you're not letting it get transferred to your character, doesn't mean that it's as fun. I love surprises and plot twists, and that is something that gets harder to do as more people share the DM seat. One way to still have this is to have a bunch of bad guys, each with their own separate plots, each run by a different DM. Then again, maybe the big reveal isn't as important.
Another way, if you're all clever at this sort of thing and everyone is cool with it, is to retcon a little when you need. Anything that happened, happened, but you might try something like "Okay, so the first time it turned out Jim Blackhat was a sorcerer. Then next time in turned out he was also a redcourt, but he had been trying to keep it a secret! But now, in his secret lair, you find the true source of his power- He's been breaking the seventh law!" Basically, add your own twist every time the DM seat changes hands. Just be careful this doesn't get too tangled and confusing, and remember not to change anything that actually verifiable happened. Blackhat fell off a cliff, but survived and is back for revenge is probably fine, we shot Blackhat in the chest at point blank but he was a vampire and regenerated might be, but around the time the players burn the body and douse the unmarked body in holy water, he does not come back. That's why evil villains have apprentices anyway!
Third, all characters get access to all milestones, even if they slept through the whole thing. That sounds a little obvious to some, and wrong and unfair to others, but especially if there are a couple of significant milestones or even one major milestone, you need to give it to all characters. Playing more than a few skillpoints behind, or even one refresh behind, is an irritating shift in power and agency. Your mileage may vary on that, but it's not hard to come up with a story to justify it if you need.
In the end, it works out pretty well. Just realize that you are giving up a certain level of genuine surprise and twist to the story, and the rotating cast will feel a little different than the same tight nit team every time.
Best Answer
The following is mostly anecdotal, but: It all depends on how big the story is going to get and what the mood is.
My party started out with five refresh, 20 skill points, and a skill cap of 4. A focused practitioner, a wereform, and a pair of minor talents. Around 30 sessions later, we concluded the campaign, at six refresh (Well, the very last session was at seven) and 35 skill points, cap 5. I've also run a few short lived games (One scenario, or two to three sessions) with high refresh characters. One time we even threw it up to 13 refresh! (and man, and an explosion that was! Haven't seen that much accidental playerdeath since the last time we ran Paranoia!) That said, those high refresh games actually had less than the recommended amount of skills.
Thing is, skills and refresh actually do completely different things in Dresden. Skills give you a way to model close, tense, mortal conflict. As long as the skill cap stays at 5 or below, the players are humans amid the world of the supernatural, and they can see the fantastic things around them, and run and hide and try to find some weakness to the horrible things outside. Every conflict probably has people walking away with consequences, and not small ones. The game has a bit of grit to it, is what I'm saying. If you've ever played shadowrun or a really good game of old World of Darkness, you know the mood I'm talking about. The more skills there are in play, the more the game is about skill. (I know, it took me a while to figure out that! Slow learner.)
The more refresh there is in play, the more the world tends to look like a story book. Strange powers become commonplace. Crazy random happenstance happens whenever it's convenient as players make declarations and the world of magic is impossible to ignore. Since skills are less powerful here, players often tend to use magic to solve problems that mortal means probably could have accomplished. A game of Exalted or new World of Darkness probably feels a little like this.
Don't think of skills and refresh as equal pieces of a 'levelup.' Think of the Skill/Refresh dynamic as a slider between mortal and supernatural, or between simulationist and storybook. The levels in the book are your halfway markers (actually, I suspect that those are the settings the authors and a majority of playtesters liked best) as apposed to a word of god ratio. Adjust them as you and your players like, to suit the tone of the game you're trying to run. Most games are built around one mood, and the system aids that. FATE isn't infinity flexible, but it can range pretty far if you tweak it a bit.