OK, here's the design I've come up with.
Here's a circuit description.
U1A debounces SW1, the RUN/PAUSE pushbutton. Its output goes high while the button is being pressed. This signal clocks U3A which is wired as a toggle flip-flop. It controls whether the circuit counts or not, by controlling the gating of the 100 Hz clock in U2B.
U1B debounces SW2, the RESET pushbutton, and provides a pulse at reset. Its output is named +CLEAR and it clears all the counters and resets U3A, so counting stops when the circuit is reset.
U4 implements a crystal-controlled oscillator that runs at 32000 Hz as set by X1. This frequency is divided by cascaded divide-by-two stages implemented as toggle flip-flops. When the count in the flip-flop array reaches 10100000 binary (Q7 and Q5 high), U1C's output goes low and U1D's output goes high, resetting the count immediately. This causes U4 to divide by 160 (128+32), producing an output frequency of 200 Hz on its Q7 output, which is divided by U3B, also wired as a toggle flip-flop, to give a 100 Hz squarewave with exactly 50% duty cycle.
The 100 Hz clock must have a 50% duty cycle for consistent timing measurement, because it is gated with the RUN signal which changes asynchronously according to button-presses.
U5 (MC14553) implements the hundredths of seconds counter. It is clocked from the gated 100 Hz signal and is held reset when +RESET is high. The internal multiplex oscillator is disabled (pin 4 is tied low) so the device always asserts the value in its bottom digit on its four outputs. It acts as a 4-bit BCD up-counter. The MC14553 advances on falling edges on its CLK input pin.
These outputs drive U10, which contains a 4-bit decoder that translates the BCD value into an appropriate display pattern, and displays it. The LEDs and logic are both powered from the 5V VCC rail. Decimal points are not used. Blanking is not used. The internal latch is permanently set to transparent so the device constantly responds to its D0~3 inputs.
U5 operates as a decade counter. When its digit wraps from 9 to 0, a falling edge will occur on its Q3 output. This provides the clock to U6, which implements the tenths of seconds digit in the same way, and clocks U7, which implements the seconds digit in the same way, and clocks U8, which implements the tens of seconds.
The tens of seconds digit needs to count 0, 1, 2, 3, 4, 5, 0, 1, 2, ... so U8's MR (master reset) input has gating that detects when it reaches 6 (0110 binary) and immediately resets the count to zero. This does not affect the other counters, which are reset only by the +CLEAR signal.
When U8's Q1 and Q2 outputs are both high, indicating a count of 6, U2C's inputs are both high and its output goes low. This forces U2D's output high, which resets the count immediately (after the propagation delays in U8), causing U2C's output to return high and removing the reset from R8. U2D includes the +CLEAR in its gating; when +CLEAR is high (active), U2A's output goes low and this forces U2D's output high, resetting U8.
U8's Q2 output goes low when its digit wraps from 5 to 0. This signal is used as the clock to U9, the minutes digit. The stopwatch can count up to 9:59.99 then it resets to 0:00.00. At that time, there will be a falling edge on U9's Q3 output, but this output is not used (except by U14).
The whole circuit runs from 5V DC. The TIL311s require a 5V supply.
If you want to use a 555 for your timebase, delete U4 and tie the inputs of U1C and U1D low or high (doesn't matter which), and provide a 200 Hz clock into U3 pin 11.