What considerations should I have when selecting a microcontroller for fast read and throughput?
I'd argue what you're looking for is very likely not what most people would consider a microcontroller¹, but a proper applications processor, which runs a somewhat usable general purpose OS. 

The ca. 70 MBit/s (=500MB/60s) throughput is, at any rate, not manageable by a CPU without heavy DMA usage. Notice that you can't likely put that much data rate through WiFi, so that's where your whole application breaks. You'll need at least Fast ethernet to do that. Since WiFi happens on a shared medium in an unlicensed band, it's not going to be reliably fully available for that rate, so I think you should rule that out. Even Fast Ethernet is a bit thin, so I'd go for Gigabit Ethernet straight from the start (this is 2017, after all, not 1997).

There's embedded processors that have GMII2 interfaces for Gigabit PHYs, but they're typically not MCU, but CPUs with external RAM.

There's absolutely no reason to design such a device yourself – go for one of the many, many single-board computers that have proper Gigabit Ethernet. I like the Odroids.

The point I make regarding "proper" OSes is that you want things that are typical of fully blown operating systems:


a flexible file system stack (notice that FAT32 can't do files > 4GB, so it might be a bad choice to start restricting yourself to)
A high-performance networking stack that supports network interfaces with DMA and interrupt coalescing (it's really no fun if your processor is stuck with handling hardware interrupts that you know will happen, anyway).
easy multitasking (transmission of data should be independent of reading from storage)
flexible buffering
Very very loose definition of latency restraints – you don't need what anyone would consider real-time.


Thus, go for a cheap embedded Single-Board-Computer running a general purpose Linux, if you ask me.


¹ definitions vary, but it's generally agreed that a microcontroller differs from a microprocessor at least in integrating the RAM in the same package)