I'm glad that this Vet chose to not do a blood-draw if they weren't comfortable doing it...It's no different taking blood from a Budgie than it is any other parrot, except that you can't draw nearly as much from a Budgie as you can say a Macaw or Cockatoo of course...But if they don't feel comfortable doing it then just say no, lol...
***Something I didn't ask you earlier that may be a very important piece of the puzzle here...What is your housing set-up with your Budgies, and how many do you have total? And do they all live together, or do some live together, what is their set-up and flock-dynamic like?
The reason I'm asking is that it's not at all uncommon for people who own multiple Budgies to keep them all together in an "Aviary" of sorts, either a very large flight cage or an actual indoor-aviary. Either way, when you have multiple Budgies living together in the same 'territory" there are all kinds of weird flock-dynamics that develop, like dominant vs. submissive, male vs. male, male vs. female, female vs. female, etc., and then of course hormones come into play as well, especially if some or all of them start or have been mating with each other, as Budgies are not at all monagomous when they are living in a "Community" flock setting. So if you have more than one Budgie living with the ones that you're finding over-preeing/plucking/bald-patches, etc., then it's very likely that it either isn't them plucking themselves all the time, but rather they are plucking each other, over-preening each other, fighting with each other (even if you're not seeing it), etc. This is very common with people who purposely practice "Community Breeding" with Budgies, and even if you're not purposely wanting them to breed, when multiple Budgies are put together to live inside of the same space, with different sexes living together etc., then this can happen...
So while they may have come to you with some malnutrition issues that are causing some feather follicle and overall feather health issues, if you adopt a new Budgie and put it in the same space with other Budgies, and even if you have had the same Budgies living together in the same space for a long time, this usually does happen to at least some extent in probably 90%+ of the time...So something to think about as far as possibly separating the ones who are showing any feather issues or bald patches into their own, individual cages to see if it makes a difference and stops the issues, giving it at least a month to see what the end-result will be. You can keep their individual cages next to each other so they can still see each other and talk to each other, but they won't be able to exert dominance or aggression on each other...You'll have your answer if in month none of your Budgies have any of these issues at all...