As smooth as it ever does...
The Disney sites are known for their instability and sloooooowness.
Really, though, even though it seems like you will be battling raging hordes of thousands of people to get your ADRs done at 6am, the truth is that there will likely be relatively little traffic around at that hour. The only ones you should be stressed about would be things like breakfast/lunch at CRT or possibly dinner at Le Cellier. I don't try for either of those, so I can't tell you how easy it might be to beat the masses to those slots, but for the most part I wouldn't stress out over the general experience/time it takes to book your list.
The system automatically searches an hour before and an hour after your requested times to present you with the closest possible matches. For example, if you put in for a table at Crystal Palace for 2 adults and 2 children at 9:45am on October 23rd, it will automatically search all available spots from 8:45am to 10:45 am, and present you the one or two closest options to 9:45am.
It will also automatically present four or five alternative restaurants, matched for how close they are to your original choice geographically and/or how closely the type or cuisine matches your original choice. So if by some odd chance of fate one of your original choices is not available at 6:07am on your 190th day before arrival, you can see what is available in the area at the same timeframe.