It is illogical to suggest that Burgess is not a quality player - or that Australia could disregard his experience - accepting the fact that I do not see how he would be eligible for this series (then again, I also don't see how a wallaby contender could conceive of signing an overseas contract that excludes him from a once-in-a-career experience like a Lions Tour, and then want to come back the very next season)...
(... likewise, I hope Carl Heyman enjoys having the dollars in the bank, enough to make up for never having a World Cup winners Medal).
I have to say that I never found Burgess' play particularly interesting or compelling in the way Genia has both these qualities (unimaginative might be the word I am looking for) - and was kind of relieved when he left, because it opened up a spot for new blood to emerge... I would opt for White as the back-up Half anytime - and perhaps the Lions Bench is not a bad place for him to mature - as any 2nd string half-back is likely going to see Very Little Game Time on this 3-match tour... The Championship however, might see some opportunity to give Will Genia some rest - and avoid further injury...
When talking about Half Backs, Australians have to speak in the context of Nick Farr-Jones, Will Genia, and (the early-vintage) Gregan... very few of the challengers in the current bunch approach even close to this level - if I were a selector, I would already be looking over them, to the next group coming through...
At other times in our history, the Wallabies have been seriously challenged for talent, at Scrum Half (Slattery, Cox & Catchpole being notable exceptions)...
