well if your 9 speed drivetrain is bascally functional but just needs some new rings/chain cables, why burn your money on a shiney 1x10 job without knowing if you'll like it, why not experiment with the 9 speed kit to at least give you an idea?
The top ratio on that 1x10 setup is roughly equivalent to what a 22t granny ring and a 25t sprocket would produce on your current 3 x 9 drive, so that's about your 3rd from easiest gear in the granny ring?
one way to simulate this would be to fit an 11-25 road cassette to your current drive train and refuse to use the big ring, that way you couldn't go lower or higher than that 1 x 10 you proposed could manage, it would of course be horrible to use and would work in silly fine increments but it's a thought.
Why not simply give 2x9 a try and see how you like that, maybe try 36/22 with an 11-32, broader range than any 1xn could manage, something to be said for that IMO, all you need do then is omit the big ring, set the front shifter so middle is top, bottom is middle and inner is nowt and set the F/mech top stop further in, this setup works for a fair few people.
Or perhaps a quick and dirty 1x9 experiment 32 / 11-32, take that front shifter/cable off, get rid of the extra rings (maybe keep the granny to catch inside drops set the mech on middle position with limit screws, shorten the chain if you like and your away, not "Perfect" but a reasonable indicator. If you can get by with that for a while, then a 1x10 will simply give you one more easier gear..
It's all relatively simple/cheap to reverse these setups is you don't like them and then you have a better idea how you'd get on with a 1x10 or 2xN without spunking £100+
All IMO of course...