I play on default and I don't find it difficult at all to melee and jump. I can do both in the blink of an eye if need be.
The thing about bumper jumper is that it means you can melee and jump without removing your thumbs from either stick. You can jump whilst still having complete freedom to look around, meaning you can guide your jumps more closely at the beginning and end, making them much more precise. It also means you can look around more freely during the jump, making aiming and fighting in the air a more reasonable proposition. As far as melee goes, its alot easier to whack someone when you don't have to remove your thumb from the look-stick to do so. It makes those quick sweeps and jumping melees so much easier, whilst also improving your chances with those quick reaction melee and assassination opportunities. The time lapse in look ability given by removing your thumb to jump/melee may seem like a rediculously small amount of time, but when I switched to bumper jumper I noticed a seriously massive change to my play for the better. Once I got used to it that is, which took about a week of casual play.
Bumper jumper has the advantage of all the combat functions being at the top, so if you play using your middle and index finger, it's amazing. As for us regular pointer finger only users, it's not to make jumping or melee faster, but to keep your fingers on the sticks at all times during combat. This allows more efficient stafing, jump control, etc. If you're used to the default, I advise trying it for a few matches. Default is fine for average players and many pros as well, so bumper jumper doesn't give that much of a help. It's mostly personal preference.
Good points, and I forgot to mention the whole index and middle finger thing. Its how I play as well and it really helps, melee becomes much more useable in the quickest of circumstances. Whilst you're right that it is, in the end, down to personal preference and its hardly going to revolutionise every aspect of your play style, I'd say the advantages in many circumstances are clear and getting used to bumper jumper is definitely worth tying properly.
The pro's that play on default(including Walshy) generally use the "claw" grip on the controller where they press the face buttons with their index finger, and that way don't have to take their thumbs off. Taking your thumb off your aiming stick to jump or beatdown is very detrimental in competitive play, where that last fraction of a second jump might save you from getting headshotted, but you need to be able to tilt down at the same time and headshot the other guy. Here's Walshy doing the claw, check out where his index finger is. Looks uncomfortable, and I can attest that it is, having played with it before switching to Bumper Jumper.
Bumper Jumper, with default sensitivity. I'm working my sensitivity up now though. When I started bumper jumper I put my sensitivity put up too. It didn't go well.
Something that is really funny to try for right-hander's is Southpaw, with the Legendary Southpaw Stick Layout.