Only two problems. Developers are now incapable of releasing bug free software due to greedy publishers like EA and Activision that just want to rush something out for a quick holiday buck even if it's not ready. Profit now, maybe patch later if we feel like it is an industry standard now.
The same driving force behind that mentality will also fight with every last breath against a media format that doesn't allow their day zero DLC practices and selling avatar clothing for $10.
So there will still need to be some kind of local storage device.
Also keep in mind SD card type memory cards are serial and block access. They cannot be directly accessed by a CPU real time, which now days we are talking 64-512 bit buses and beyond (integrated GPUs etc) and multi GHz speeds. I don't know of any SD card that can provide nanosecond random access and 50+ GB/s.
This console will still need to operate the way a current system does, with lots of RAM to load assets into, treating any card formats as a block storage device just like any mechancial media, and a mass storage device for patches, OS updates, save games, parallel loading of data, etc so it won't truly be like old school cartrides that plugged right into the CPUs A/D lines.
With todays speeds and voltages, with path lengths and stuff being so tight due to capacitance, cross talk, clock skew, etc, I seriously doubt you could get away with a "real" cartridge format (eg: parallel CPU A/D lines in the cart slot). Everything is going to a LVDS serial formats for this reason (SPI, SATA, PCI-E, HDMI, HT, QPI, etc). A CPU at 3 GHz you could have glitches if the cart wasn't inserted with quite the same exact force and distance side to side. Ever see those tiny squiggly traces on a modern PCB to keep the traces equal length? It's that sensitive. Just getting signals to arrive at the same time in different parts of a 1 cm square silicon chip is a science all by itself, let alone through some large rugged exposed cartridge port that somebody has handled with sticky fingers.
<-very experienced low level software programmer with some basic hardware knowledge including FPGAs, VHDL, etc.
This is for my fellow SNES lovers:
lda #$00
pha
pld
lda #$07
sta $2105
nop