I'm a professional forecaster (i.e. getting paid for it) at a large e-commerce company. We have extensive experience with Prophet and a host of other approaches (all the traditional models in Hyndman's book/R package, some scattered LSTM/NN implementations). Here's my quick take (the article is a lot more extensive than the median blogpost, and likely warrants a more extensive study than I have time for right now.)

Prophet main claims ("Get a reasonable forecast on messy data with no manual effort. Prophet is robust to outliers, missing data, and dramatic changes in your time series.") are surely exaggerated. As the article shows, time series come in many different shapes, and many of them are not handled properly.
It deals well with distant-past or middle-of-the-sample outliers, but not with recent outliers. It cannot deal with level changes (as opposed to trend/slope changes). None of this should be a surprise if you take some time to understand the underlying model, which unlike most neural nets is very easily to completely understand and visualise: it's really a linear regression model with fixed-frequency periodic components (for yearly seasonality and weekly seasonality) and a somewhat-flexible piecewise-linear trend. The strong assumption that the trend is continuous (with flexible slopes that pivot around a grid of trend breakpoints, which are trimmed by regularisation) accounts for most of the cases where the forecasts are clearly wrong.

That said, it does occupy a bit of a sweet spot in commercial forecasting applications. It it's largely tuned for a few years of daily data with strong and regular weekly and yearly seasonalities (and known holidays), or a few weeks/months of intraday and weekday seasonalities. Such series are abundant in commerce, but a bit of a weak spot for the traditional ARIMA and seasonal exponential smoothers in Hyndman's R package. These tended to be tuned on monthly or quarterly data, where Prophet often performs worse. In our experience, for multiple years of daily commercial-activity data, there are no automated approaches that easily outperform Prophet. You can get pretty similar (or slightly better) results with Hyndman's TBATS model if you choose the periodicities properly (not surprising, as the underlying trend-season-weekday model is pretty similar as Prophet, but a bit more sophisticated). Some easy win for the Prophet devs are probably to incorporate a Box-Cox step in the model, and a sort-term ARMA error correction, then the model really resembles TBATS. You can usually get better results with NNs that are a bit more tuned to the dataset. But if you know nothing a priori about the data except that it's a few years of sales data, your fancy NN will probably resemble Prophet's trend-season-weekday model anyway.

All of these assume that we're trying to forecast any time series' future only from its own past. If you want to predict (multiple) time series using multiple series as input/predictors, that's a whole new level of difficulty. I don't know of a good automatic/fast/scalable approach that properly guards against overfitting. Good results for multiple-input forecasting approaches probably requires some amount of non-scalable "domain knowledge".