I teach Year 6 and, at primary, most percentage questions are in 5% or 10% jumps and you wouldn't tend to get answers to two decimal places. Therefore, I agree that (89 x 39)/100 is probably the most efficient method in this case and the mental method is potentially quite complicated.
However, as you say, irvine, the real aim is to expand thinking. The point of the new curriculum especially (with its non-calculator approach to primary maths) is to develop a conceptual understanding of number as opposed to just techniques for dealing with particular problems.
If I were working it through with a Y6 student and trying to make it less complicated... well, first off, I'd start with 30% of 350, 85% of 470 and 40% of 3 because these require fewer steps.
But sticking to your question, I'd make sure they use jottings and their wider maths knowledge. So jot down:
10% of 89 = 8.9
1% of 89 = 0.89
Now, to find 40% do 8.9 x 4. You can do this mentally if you know that 8.9 is the same as 9 - 0.1. So four times that is 36 - 0.4 which is 35.6.
So I'd jot: 40% of 89 = 35.6.
Now you need to take 1% from 40% which means calculating 35.6 - 0.89 .
Because we're going into decimal places and we're subtracting, I'd probably draw a blank number line as a prompt -- this is what children use lower down the school and, although you can do it without, it's still a useful prompt to iron out misconceptions such as whether you're counting up or down.
So I know -0.89 is the same as -1 and then +0.11 (which I know because 11 is the complement to 100 of 89 and therefore 0.11 is the complement to 1.00 of 0.89).
35.6 - 1 = 34.6 -- that'll take me to the left-hand side of the number line. Then add 0.11 (a jump part-way to the right on the number line) to get 34.71.
What I'm trying to show is that, to work through this question, there's a lot of conceptual understanding needed. The child needs to know how to tackle percentages. How to subtract decimal numbers which are to different decimal places. How to simplify expressions to make them easier to calculate. Knowing complements to 100 in different powers of 10. And not to be fazed by needing a few steps to get an answer.
Children should have all this by Year 6 if they're working at the standard. And none of the individual calculations are very complex. So if the child is missing a step, you can work out which one they're missing.
And if you've found something they're missing -- THAT'S the gap you plug first because it's obviously a more fundamental gap than calculating percentages.
By the way, once you've shown both methods, the extension of this would be to ask the child to use their reasoning skills to compare the two approaches. Is (89 x 39)/100 the same as the above method? Prove it -- check. What's similar and what's different between the methods? How do you know? Which is more efficient and why? Is it the one with the fewest steps or the simplest calculations along the way? Which would be the best method to use with a calculator? Which would be the best method if you were estimating in your head?
It's more involved than just having the one strategy but it's less about teaching "the way to do it" and more about developing a deeper understanding of how to manipulate numbers in lots of different ways to reason and problem-solve.