A 555 timer is not inaccurate, it is around 1 in 100000 error, plus any change that might happen in the components. A 10 second time is a little long, and electrolytic or tantalum capacitors should be avoided. Also hi-K ceramic capacitors and cheap carbon resistors. However metal film resistors are 50ppm per degree C temperature change, or better. Also chose capacitors with low temperature co-efficient and low dielectric absorption.
It can be used as a faster (more accurate) clock with say 1 second or so with a simple digital counter chip like the 4017 (decade counter) to get around 10 seconds.
With the same proviso on components, an op-amp with low bias current can be used for longer times. See the LT1012 for an example. Use a precision voltage reference to drive an integrator and a precision voltage comparator.
When it comes to analogue components the simulator may not model capacitors very well with parameters like dielectric absorption. Also a simulation needs to be done at different temperatures, assuming the components are able to have temperature coefficients, and different voltages. These are the two most likely causes of changes. Another is aging. Adjustment trimmers may be another error source, as they are less stable in general, when compared to fixed components. To get accurate results using a timer, some sort of calibration procedure is needed, and this may well turn out to be the biggest error.
A 4060 IC has a RC oscillator and divider chain. There are various timer ICs other than the 555 which work in different ways, but these might not be readily available.
Analogue wall clocks use a tuning fork crystal oscillator at 2^15 Hz = 32767 Hz. This is accurate to around 1:100,000 usually. A counter circuit divides by 2^15 to produce 1 second pulses. These drive the stepper motor. However they could be used to drive a 4017 counter to get up to 10 seconds, or other counters for different times, just as with the 555 timer circuit mentioned above.
You could look these things up, search images for "op-amp timer circuit for example". The link below is an example of an op-amp timer circuit, with a second input that integrates the light received from a photo-diode. Change the capacitors and resistors for different voltages and times. Use an LT1012, or maybe an LM308 or LT1055/6, LF355/6 or LF411 (for low bias current). Ignore the 300K resistor and 680pF capacitor. With a 10 megohm input resistor, a 1uF timing capacitor and a 1V reference it will ramp at a rate of 1V in 10 seconds, and so trip at 1V in 10 seconds after a reset. You might need to implement a voltage controlled switch for the reset in a simulation, and figure out a way to do this in hardware too (a relay maybe?). There are diodes shown in the circuit that are to provide reference voltages (2CP12 and 2CV I think), replace then with accurate reference diodes. There could be an advantage in using the same voltage reference too, so the two reference track. Look up voltage comparator, schmidt trigger, op-amp integrator for more info.