Continuous sample size determination based on the control group I am curious if and where the following reasoning breaks down.
Traditionally, sample size determination is done as part of the design phase. To this end, one has to have an understanding of the baseline performance, such as the mean and standard deviation of the metric in question. One might use recent historical data to obtain reasonable estimates.
Suppose we are not necessarily interested in knowing the minimal sample size ahead of time and simply launch the experiment. The control group is the baseline. Each day, we take the statistics of the control group and perform sample size determination. With each new day passed, we get a better and better understanding of when to stop.
Apart from the fact that one might launch an experiment that is doomed to fail, are there any other problems with the above logic?
Clarification 1: It should be noted that this question is not about whether it is sound to check some p-value on a daily basis and stop whenever it goes below a predefined level. The inadequacy of this procedure is well understood.
Clarification 2: Here we are not assuming any clinical or otherwise setting with serious implications and ethical concerns. Rather, one can imagine something trivial, such as an experiment with a webpage where the color of a button is subject to change.
Clarification 3: The question describes a specific stopping rule: every day, run a power analysis using the statistics of the data for the control group collected up to today and terminate the experiment as soon as the actual sample size exceeds the one given by the power analysis, potentially skipping a few first days to avoid pathological cases.

A priori sample size calculations are only sensible if there is a fixed cost to sampling
This is an excellent question, and it raises some common-sense issues that are commonly misunderstood in sampling theory.  When we are considering sample size questions, the first thing we need to ask ourselves is what (if any) impediment, inconvenience, cost, etc., there is to just collecting all the data there is on the matter of interest.  If there is no cost to this (construed broadly as including time, money, etc.) then we might as well go out and collect all possible data on the matter and then we don't need statistical inference procedures at all.  Consequently, the idea of using a limited sample size that is less than what is possible only arises as a sensible procedure when there is some cost to sampling.
Once we have identified the costs of sampling (again, construed broadly to include more than just money), we need to decide if there are any fixed costs that accrue when we conduct a sampling "excursion", or not.  If there are no fixed costs then there is no need to limit our data collection by any a priori calculation --- we can just collect some data, decide if it is enough, collect some more, decide if it is enough, and so on.  So in this case, we can indeed to something like what you describe; sampling without any a prioiri sample size calculation and then stopping when we have enough data for the inferences we want to make.  (We need to be a bit careful with this method, because certain kinds of "stopping rules" combined with particular statistical procedures can bias analysis, but assuming we are cognisant of this, we should be able to formulate an appropriate "stopping rule" and stop when we have the data we need.)  Various statistical control processes operate in exactly this way --- they collect data as it is generated and update inferences with each new data point.
If there are fixed costs to sampling, but the fixed costs are quite low relative to the unit costs, it might still be useful to sample over more than one "excursion".  In such a case we might make an initial a priori sample size calculation for an initial "pilot" set of data, then use this data to make an updated sample size calculation for the next set of data, and so on.  In this case there might be a number of data collection excursions, but each will involve a sample size calculation.  This is also somewhat similar to what you describe.  Finally, if the fixed costs of sampling are quite substantial relative to the unit costs, then the "traditional" method of using only one sampling "excursion" is probably going to be optimal.  In this case the sample size is fully determined by a single a priori calculation.
So, in terms of where your reasoning breaks down, it is only in the possibly exaggerated claim that "[t]raditionally, sample size determination is done as part of the design phase."  This is only true if there are substantial fixed costs to sampling that make it desirable to sample "in one go" and thereby necessitate an a priori computation of the entire sample size.  Your observation that it can be sensible to collect data "as you go" is perfectly legitimate (under appropriate circumstances), and it is actually an underappreciated principle of sampling theory.
