Why use a 1-bit ADC in a Sigma Delta Modulator?
First of all, because it's easy to build a 1-bit ADC. It's a comparator. It's literally the easiest ADC you can build. The $\Delta\Sigma$ ADC was invented (or, rather, published) in 1962¹ !
The 2-bit ADC is more than twice as complex as that,  you need some window decision: so if you have the choice of making your 1-bit ADC run faster or building a somewhat exact 2-bit ADC, there's a solid chance you'd go for the 1-bit ADC, simply because there's less analog semiconductor design to do, and, more importantly, to go wrong!
Imagine the 1-bit ADC doing a sign decision (&quot;is the analog voltage &gt; 0V?&quot;). No matter how you scale the analog voltage (multiply it with a factor $\alpha$), the result will always be the same.
Thus, the 1-bit ADC-based $\Delta\Sigma$-ADC can be scaled to fit any range simply by changing (scaling) the threshold (which effectively is the analog part of your 1-bit ADC) in the forward chain (picture from wikipedia):

It's absolutely not obvious how you'd change above diagram to fit a 2-bit ADC! But, above diagram is very close to the original publication¹, so abstraction yields that the threshold is just a single-bit quantizer.
However, it's been done, there's multiple patents² covering higher-quantization  methods. You'd have to limit the bandwidth of the forward chain³, and, depending on what you want to achieve (class-D amplifier, faster ADC, lower power consumption…), you'd then have a bit of clever logic that translates the ADC output into different kinds of feedback &quot;pulses&quot; and different kinds of counter increments.
So, your original claim &quot;we don't use more than one bit in the quantization within the loop of a $\Delta\Sigma$ ADC is wrong; it is only right for the very classical implementation of the original inventors.
To totally disprove your point: There's a lot of $\Delta\Sigma$ ADCs on the market that actually do higher-order quantization; for example, the Analog Devices AD9267 uses a nine-level quantizer⁴ (3.something bit, which was a pretty impressive feat on its own to integrate, considering the speed of 640 MS/s of that quantizer).

¹  Inose, H., Yasuda, Y. and Murakami, J., 1962. A telemetering system by Code Modulation – $\Delta$-$\Sigma$ Modulation. IRE Transactions on Space Electronics and Telemetry, (3), pp.204-209.
² WO2008028142, which already cites multi-bit feedback as &quot;prior art&quot;.
³ otherwise, you'd always only get the lowest ADC output before you could get any of the higher ADC outputs, and with infinite bandwidth, that would mean you'd already emit a pulse to &quot;reset&quot; the integrator, and you'd never use any but 1-bit of the ADC. That bandwidth-limiting typically happens &quot;involuntarily&quot; because electronics are limited by physics.
⁴ AD9267 Datasheet, p. 13, &quot;Theory of operation&quot;