One reason for using a bridge is because the resistances are very small. The resistances of the connecting wires etc can not be ignored, and any electrical interference like voltages induced from AC mains wiring or stray magnetic fields would cause errors.
If you build a bridge using "identical" components all mounted the same way, these errors are cancelled out.
A Wheatstone bridge circuit that had to be balanced by a human operator would be too slow to use for large numbers of gauges, and could not be used at all for measuring strains that vary with time, for example if the component is vibrating. Strain gauges can easily measure the response of a structure at frequences up to tens of kHz.
How would the errors be cancelled out?
IF there are any induced EMFs, approximately the same EMF will be induced in all the wires if they follow the same geometrical path, therefore the resultant is only the (small) difference between the EMFs, not the (bigger) total induced EMF.
The original Wheatstone bridge was invented in the 19th century, before the era of electronics. Apart from as a teaching tool (i.e. they are a circuit that you can analyse and experiment with) electronic measuring instruments have made them completely obsolete.Based on your last paragrah, are Wheatstone bridges used at all in industry?