Calling an election or referendum a waste of money is to be against democracy.
Thing is, arguably, having a by-election is also kind of undemocratic. Ipswich council has mixed member electorates, so the first and second place candidates are elected, resulting in a more proportional result than single-member electorates. If they run a by-election, though, there's only one seat to fill, so the result could end up being less proportional (and more proportional means more democratic - in many people's thinking, anyway).
I'll give an illustrative example...
Say the original election results are (let's pretend preferential voting doesn't exist, or I guess that GREEN party preferences split exactly equally for BLUE and RED):
1st - 34% - BLUE party candidate
2nd - 33% - RED party candidate
3rd - 31% - GREEN party candidate
The result is that the ward is represented by one BLUE party councillor and one RED party councillor.
Now, if the RED party councillor resigns, in the by-election, if people vote exactly the same way as in the original election, the result is that the ward will be represented by two BLUE party councillors. That'd mean in this (very contrived) example, 34% of the vote would give BLUE 100% of the seats.
I can see why you'd argue that appointing the runner up (the GREEN candidate) also isn't democratic, but in this (very contrived) example, doing that would mean that 34% of the (primary) vote gives the BLUE party 50% of the seats, and the 31% of the vote gives the GREEN party 50% of the seats. So things would end up more proportional.
Like I say, though, that example is contrived to show that result. If we were to say that GREEN party voters strongly preference the RED party, then the RED party would win the by-election, and then the result would be almost exactly as proportional. If GREEN party voters strongly preference BLUE, though, then again, we're back at two BLUE councillors.
My point is that, depending on the situation, having a by-election for one seat in a two seat electorate may result in a less proportional/democratic result than simply appointing the runner-up.
That said, I think council elections tend to be less dominated by the bigger parties, with a lot more independents, and less of predictable, party-lines voting. So maybe that affects the likelihood of either option resulting in a more or less proportional outcome...