The Description of the sine shaper give an optimum Amplitude of 3,25 * U_t. The voltage U_t is proportional to absolute temperature. This temperature dependence is typical of circuits using the nonlinear diode curve. If the temperature is reasonable stable one may live with that. Beside distortion the output amplitude will also depend on temperature a little. An improved sine shaper may use a regulated temperature for that part, or temperature dependent resistors (e.g. R24 and R19 as PT100 and PT1000 (or similar)). A current mirror with discrete transistors should use emitter resistors for more accuracy. Well coupled and matched transistors may work, but two extra resistors are easier.
One particular bad point is the output amplifier. At least this is easy to fix: feedback has to be on the other side of R30. One could replace R30 with a short and have 50 Ohms directly at the output connector.
With the measured curves, it is not clear weather the distortion is from the generator, or just from the way how the scope is connected. A poor ground connection and the poor output amplifier without termination - may produce such ringing, especially if 1:1 probes or similar are used. Such ringing also depends on layout and power supply decoupling.
The low frequency signals just show the effects of AC coupling of the scope, and the ringing is lost by the DSO - it's very likely still there, just not visible. So the bigger trouble may be due to the Scope, though not a DSO artefact, but more likely a ground problem.