I stumbled on this thread last week and was surprised the following fix didn't get mentioned: with three more resistors and one power supply connection change, the circuit can work as originally intended.
As previously noted, the LT1719CS8 output stage cannot operate between +/-5V. But it can operate 5V single-supply, including from 0V to -5V. So with the changes below, the left-hand portion of the circuit is limited to that negative span. The OPA2810 non-inverting input is lowered to -2.5V by the additional two 4.99K resistors, and this is a better choice than +2.5V due to an aberration in the input stage at that level, when the "main" input stage is switched over to the the "auxiliary" one; see Figure 18 in the datasheet as well as the specification tables on pp6-7.
With the LT1719 output voltage swing half what the circuit was originally designed for, the R1 & R3 values are cut in half to preserve the same current swings. The right half of the circuit works essentially the same as before, but because the current of R3 is biased at 0 or 1mA, whereas before it was biased at +/-0.5mA, the added 10K resistor to +5V is needed to source 0.5mA into the comparison node to compensate.
Note: the actual current through R3 and the new 10K resistor will not be static, fluctuating with the Output waveform. These current levels mentioned are the currents where it counts: near 0V when the LT1719 comparator will trip.
While this circuit will happily preform as a Triangle Generator, the LT1719 output signal should not be connected to any logic gate inputs (to, say, monitor the "clock" of the output) because it is operating from -5V to Ground.