The idle stumbling in your video is just the way that these bikes all run in stock form. If you listen closely to a lineup of FJRs you'll hear that they (mostly) "all do that".
It is caused by the mixture (intentionally) being a bit on the lean side, for emissions purposes. Even with a PCIII (and disconnected O2 sensor), unless you tweak your fuel map in the cells in the upper left corner of the map (at zero throttle and 1000-1250 rpm) it will continue to have that burble. I have actually tweaked those cells (along with others in that column) and have made my idle considerably smoother. That said, it really isn't a big problem and definitely not worth getting a PCIII just to get rid of that.
However, sensitive riders may also notice that the engine actually exhibits this same "lean stumble" whenever the engine's fuel injection goes into the closed loop mode while riding along. In addition to at idle, closed loop mode happens whenever cruising at a steady throttle setting and relatively light engine loads. The way it feels is like the bike's speed is "lurching" just a little bit, even on very smooth roads and with no winds. This was actually the phenomenon that prompted me to get a PCIII for my 1st gen.
The added bonus of having the PCIII (and disconnecting the O2 sensor) is that the ECU will never go into closed loop mode, and you can add fuel up and down the first few columns of the map where the 0% and 5% throttle openings are. You will never be cruising or riding around at these throttle settings. What they will effect is what happens when you close the throttle down as you come into a corner, and then want to roll back on the throttle mid-corner. The default ECU fuel map cuts off all fuel under trailing throttle situations, which makes the roll-on very herky jerky. Exactly what you
don't want in mid corner. Adding fuel in these columns softens that transition and makes the bike much more ride-able.
Curiously, the maps that you get from DynoJet, and most of the other fuel maps that you see around, do not make any attempt at enrichening the zero throttle column of the fuel map, instead paying attention only to 5% and greater throttle openings where cruising and acceleration happens. However, if you do diddle with the first column, whether or not your bike would pass an emissions gas test is surely in doubt. These settings are for "offroad use" only