Well you should certainly be able to do a lot better than that... why do you think you'll be stuck with that level of inaccuracy?

My admittedly limited testing has showed the probe (runout adjusted each time) to find centre of a bore to within less than 0.02mm.... a quick way to check this is to run a centre probe, take a note of the Machine co-ordinates rather than the work co-ordinates, clock the probe eg 90 degrees and run the centre probing again. note how much the machine co-orindates have changed by after it re-zeros, repeat that a few times at different clocks and the maximum variation in machine co-orindates gives you a good indication of how reliably it is finding centre.

Finding zero from probing two edges to find a corner is a slightly different kettle of fish as you would need to calibrate to take account of deflection before the probe triggers to get truly accurate results. Centre probing mostly gets rid of this issue as deflection occurs on both sides.

If machining using a jig/soft jaws, you can always ream a bore in the jig/jaw that will then serve as your zero point.

I would say with a bit of thought you should certainly be able to get two sided machining to within a very tight tolerance (in terms of repeatability - absolute accuracy is another question as you could for example be eg 0.1mm out on all dimensions but as they will all be the same then relative to each other it doesn't matter).