Modelling Unevenly Spaced Time Series
ARIMA, Exponential Smoothing and others indeed require evenly spaced sample points. As you write, you could bucketize your data (say into days), but as you also write, you would lose information. In addition, you may end up with missing values, so you would need to impute, since ARIMA is not very good at handling missing values.

One alternative, as you again write, is to feed time dummies into a regression framework. I personally do not really like categorical dummies, because this implies a sharp cutoff between neighboring categories. This is usually not very natural. So I would rather look at periodic splines with different periodicities. This approach has the advantage of dealing with your uneven sampling and also with missing values.

Be very careful about interpreting $R^2$. In-sample fit is notoriously misleading as a measure of out-of-sample forecast accuracy (see here). I would argue that this disconnect between in-sample fit and out-of-sample forecast accuracy also means that there is no connection between in-sample fit and how well a model "understood" the data, even if your interest lies not in forecasting, but only in modeling per se. My philosophy is that if you can't forecast a time series well, you haven't understood it in any meaningful sense.

Finally, don't overdo the modeling. Just from eyeballing your data, it is obvious that something happened in June, on one day in August and in September/October. I suggest you first find out what this something was and include this in your model, e.g., as explanatory variables (which you can include in ARIMAX if you want to). What happened there is obviously not seasonality.