Designing Precision Instruments: what to self-calibrate, what to guarantee by design
There is no one right answer and the right answer in any given situation will tend to change with time as different components become available  (or unavailable) and costs and various prices change. At one time 0.1% resistors were pretty expensive in smallish quantities, now they're pretty reasonable and available. A 0.01% resistor may cost $15 or so in low quantities- that's only 13 bits of accuracy. If you want to approach the possible 16-20 bits of accuracy with 24+ bit ADC converters it's going to be difficult without some kind of calibration. But you may still need the $15 resistors if you want that calibrated value to stay stable with time and temperature. 

The second option is completely practical with analog instrumentation - you just use adjustable, selectable (or selected) or trimmable components. Computers (externally) may be involved. It was once common to use computers to abrasive or laser trim resistor networks in situ to calibrate instruments. 

You really have to run through all the plausible options and see what makes sense in a given situation. If there is only a single part that affects precision and it's available in a high volume reasonable cost part, it's often best to just buy that part. For example, an ADC reference. If you only need 0.1% accuracy and you can buy a 0.08% reference that guarantees initial accuracy and has superior stability to the 0.25% part plus trimpot or plus digital gain correction then that may be the correct way to go. 

When you have done many instrumentation designs, the number of plausible options will go down as you can reject options that are not likely to be competitive at an earlier stage (one reason why experienced Engineers make a lot more money). 

You need to consider the error budget, the desired stability (and other performance characteristics such as noise) the parts cost and overall cost including amortized cost of calibration jigs, procedures, training and in-field recalibration- what happens if an EEPROM gets wiped and your wonderful +/-10ppm calibration suddenly shifts by 5% of full scale error? vs. simply testing to verify that the system with purchased parts is actually working to spec (you definitely should not attempt to omit that step!). 

If you are calibrating you can probably hit the center of the tolerance band pretty closely (well, maybe depending on how it's done- resolution of digital calibration still has to be sufficient) at the factory door, so a bit more drift with temperature and time could be allowable and still meet the spec. If you are unlucky and several parts contribute additively to the error budget then you could be near the edge and a small change would put you out of spec. In some cases, stability may be more important than absolute accuracy anyway. 

Using digital calibration you can even correct for temperature changes by putting the instrument in a chamber and calibrating it at various temperatures. This is a major PITA and is to be avoided in all but the most extreme situations, in my experience.