You're absolutely right. The answer is in the politics of the past, and somewhat the economics, which, of course, is related.
I would suggest to achieve a change warranting a stronger RAN you would need to start with Australia's move away from Britain and towards the US. That takes you back to the decision to allow US bases in Australia and, especially, participation in the Vietnam War, and then Britain's withdrawal from east of Suez.
So, how about we go to the 1963 election, where the incumbent Menzies government won an increased majority against Arthur Calwell's Labor opposition due, largely, to some own goals on the part of the latter. If the result goes the other way and Calwell wins, then we likely won't see Australian participation in Vietnam - or at least a much reduced level of participation. Subsequently there would be no need for the introduction of national service.
Now, as for the RAN, we could see a range of different decisions that could lead to where we want to go. HMAS Sydney could be paid off given she wouldn't be needed as a fast troop transport. That saves money. Until the decision was reversed in 1963, the Menzies government had decided to convert HMAS Melbourne to an ASW carrier with Wessex helicopters and no fixed-wing aircraft. The hypothetical Calwell government may decide to go ahead with that plan.
I can hear people reading this and thinking you've just gone two steps backwards, which is true, but maybe that was what was needed. In the same era, the mid-'60s, you have the Royal Navy fighting for its next major ships, with the "through-deck cruiser" being developed, and you have the predecessor to the Harrier making an appearance. So maybe the Royal Australian Navy has a similar philosophy, seeing submarines as its greatest threat, and takes a keen interest in what's happening in the Mother Country.
By the mid-'70s, irrespective of which party is in government, it is conceivable - particularly with the UK's withdrawal from east of Suez and the outcome of the Vietnam War, but maybe added with some hypothetical increase in Soviet naval activity in the region (maybe another South-East Asian nation buddies up with the USSR) - that Australia decides to order a ship of the Invincible Class and some Sea Harriers to increase the ASW capability provided by HMAS Melbourne. Then, a mere few years later, well before that ship is even delivered, the government decides it might as well get a second to replace the ageing carrier.
There would, of course, be a whole range of other decisions that would go differently. The Perth Class destroyers were all ordered before the 1963 election so nothing changes there. But the Adelaide Class (Oliver Hazard Perry) frigates might be butterflied away. Perhaps four Leander Class frigates are ordered instead - converted during construction or shortly after perhaps for Sea Wolf. Aside from the Navy, it is interesting that Calwell went into that 1963 election promising an earlier replacement for the Canberra, which likely means the F-111 doesn't see Australian service and you have either the Phantom or the Vigilante instead, and maybe, if the former, an improved AAR capability.
Back to the RAN, I could see a 1985 surface fleet being: 2 x Invincible Class carriers, 3 x Perth Class destroyers, 4 x Leander Class frigates, 6 x River Class destroyer escorts.
The carriers won't need replacing till 2010-2015 or so, and no doubt there'd be immense debate surrounding such a capability. Perhaps the new ships - if they are acquired - would have a greater emphasis on a secondary role of amphibious operations, so a modified version of the two Canberra LHDs we've ended up with, but perhaps too the Sea Harriers might have been replaced by Harrier IIs and a commitment made to acquiring F-35Bs too.