Hi,
I use three systems to measure for the lower frequencies.
An audio Precision measurement set.
Siglent SDG 1032X
Qaunt Asylum 403
If you so want to start doing measurements on audio equipment and the THD is not super important then the Siglent SDG 1032X is the first thing you could buy.
The THD is a little under 0.1% and for most Audio measurements that is sufficient.
But more important in my eyes is the flawless pulse response of the Siglent SDG 1032X, there are no abbarations if you are terminated with 50 Ohm and you do not go higher than 1Vpp,
the square wave is then perfect.
This is what the square wave pulse looks like, no abberations!

This is a measurement also done with the Siglent SDG 10132X but measured with the Quant Asylum 403.
What was already pointed out by another forum user, this measurement instrument is better suited for distortion measurements and noise behavior.
The generator of the Quant Asylum also has a nice low distortion.
This picture shows the distortion of the SDG 1032X along with some random other signals always present with DDS generators like this Sigelent SDG 1032X.
This is a 1KHz Sine generator Loaded with 50 Ohm.

Now which is best for the topic starter, actually a Siglent SDG 1032X and a Quant Asylum :-)
It is better not to go any cheaper in terms of function generator than the Siglent SDG 1032X which I have had about four years of experience with now.
It is together on the measurement shelf with the SDG 2042X and also a Rigol DG4162.
But usually the first one that comes on to do measurements is the SDG 10132X, because I often need a clean pulse signal and if I stay below 1Vtt at 50 Ohm load I don't have to worry about my signal.
Above this signal level a small riple is visible just like with other function generators.
Also remember that with a function generator you can do much more than pushing a 1KHZ signal into and amplifier?
You can also tune radios with it, I test power transistors with it, the generator drives the base and with an external power supply and collector resistor I then do some basic measurements.
The function generator can also be used as a DC source, keep in mind that the output resistance is then 50 or 25 Ohms when closed.
There are also so many possibilities with a modern function generator, don't forget to take that into consideration.
Kind regards,
Bram