The advantage of a microcontroller in your project is that it'll replace a lot of other logic you'd need to "count" the time and render it on an LED display. Instead of multiple counter ICs, BCD-to-7-segment decoder ICs (or 26 diodes as you mention), and decade counters and drivers for multiplexing the display, you can do it all with one microcontroller.
To retain time during a power outage you'll need some sort of backup power source, such as a battery, and a clock source to keep it "ticking". If you use the DS3234 it'll provide the clock source. If you use the DS3234 as the clock and use SPI to a micro to drive the display, your backup power wouldn't need to power the micro at all, just the DS3234. You'll lose the display during outages in this case, but the battery will last much longer.
You may not even need the DS3234. A mid range PIC can drive its Timer1 register with 32.768 KHz watch crystal directly, and can run even in sleep mode during power outages.
If you plan on using the mains frequency as a clock source, and power outages are rare, you could probably get by with an RC oscillator as a backup clock source, though a crystal oscillator will be much more accurate.
If you really want to get creative you could add a GPS module and use that to set the clock automatically.
The ULN2803 is a Godsend. I always keep these on hand. They're great for 7-segment displays as long as they're common anode.