This sounds like you're in an explorative stadium of your analysis and want to get a feel for the data. I think the idea of grouping time series with a similar trend might be a great idea. It's a bit of a mixture of supervised learning (you have some idea of what you want to see grouped together, vague as it may currently be) and unsupervised learning (you want the algorithm to bring order into the data in a way that you don't prescribe completely).
In this setting, it's probably best to use supervised methods to generate features related to trend, for each of your time series. There are several time series methods available for this, starting with, but not limited to, good old ARIMA. Then, use an unsupervised method to cluster your time series, based on the trend-related features you created, and maybe some additional features of your choice.
I would prefer k-means or k-medoids over t-SNE for this, as the results of the latter depend on your choice of parameters in a much less transparent way than in the case of k-means/k-medoids.