Can I put the problem in the background?
Yes. It's quite common for the problem to come out naturally in the discussion of the prior art. This reflects the conventional British (and thus Australian) drafting style. In my view, if you ignored IP Australia's suggestion in this case, you would be simply following the path tread by many attorneys who also do so.
To be honest though, I can't see it makes any difference where you have the heading "summary". The text probably wouldn't change in either case. For example:
Existing motor vehicles provide a seat on which passengers can sit during operation of the device.
So here, we have given a brief discussion of the prior art (as you see it). Anything you place here is effectively admitted as being prior art, so you should be mindful not to explain too much.
SUMMARY (first possible placement)
In existing motor vehicles, during collisions, passengers would tend to be propelled from the seat through the windscreen, which many passengers find undesirable, along with leading to likely damage of the windscreen. There is therefore a need for a means which ameliorates this, or at least provides the public with a useful choice.
Here, the prior art ("existing motor vehicles") has a problem (passenger propulsion during collisions), which the present application looks to address. The solution is then set out, along with comments as to the benefits of the solution. For example:
SUMMARY (second possible placement)
In a first aspect, there is disclosed a seat comprising a belt.
On collision, the belt restrains the passenger, reducing the change of propulsion of the passenger during collision.
Here, the "first aspect" mirrors the first independent claim, and you've explained how the subject matter of claim 1 addresses the problem you identified above.
Why do this at all?
By way of background, the format that IP Australia provides on this was agreed by the IP5 (that is, USPTO, EPO, SIPO, KIPO and JPO) to align the structure of patents across the offices. In principle, by following this, no office would have an objection.
Despite the USPTO being part of the IP5, this approach is kind of anathema to US-style drafting, as Aldo noted. If your only key market is the US, it might be sensible to jettison this approach. It really depends what your plans are.
By following this approach, you are essentially setting out your argument for an inventive step (though without explicitly arguing why it's non-obvious). This allows you to frame arguments during prosecution better, and improves your chance of a smooth prosecution. In contrast, if you don't include this, there is a risk that the examiner will see your improvement as being an obvious design variation of existing systems with no advantageous effects, and therefore lacking an inventive step.
In Australia, you have a bit of freedom with subsequent arguments if this happens. However, in stricter jurisdictions (notably the EPO), this kind of problem–solution–effect statement is extremely helpful. Without it, you're generally very limited in what you can argue.