The serial dilution issue is a separate issue with independence, not homoscedasticity. Conceptually the issue is that the calibration curve won’t “generalize” as well to out-of-sample data if you do serial dilution. Reason being the non-independence causes the errors to be lower for the correlated observations on the curve but the unknown observations you are trying to predict from are not part of this.
Indeed OLS is still unbiased with non homoscedastic errors. However, it suffers from precision issues. The molar absorptivity you estimate will be further from the “truth” on average. By the way, this issue actually gets worse the larger the experimental error is. This can be shown in an R simulation.
Michaelis Menten Kinetics is perfectly valid if you do a Normal GLM inverse link, NLS, or WLS model. And actuallt the original Lineweaver Burke paper suggests the WLS model ( weighting by 1/y4 after inverting both y and x). So indeed their original work is solid, but this part was totally missed. However, many people these days do use the Nonlinear Model pre-inversion which is also valid. In fact it can be shown that the weighted method is the first iterative step in the inverse link Normal GLM or Nonlinear LS model (which are the exact same models).