Disclosure: I currently own an '06 Rocket III Classic and an '08 R 1200 GS Adventure which follow many years of riding liter-class sport bikes almost exclusively.
If a sport bike is consistently beating an RIII off the line or through the first couple of gears the cause is either an extremely tricked sport bike (think 'busa turbo) or a fairly inept RIII pilot. But most sport bikes will begin erasing the RIII's launch advantage when the RIII hits 3rd gear and many will have caught the Triumph by the time the RIII hits 4th; from that point on up to top speed the sport bikes will easily best the Rocket. Aerodynamics prevent the RIII from performing well at high speeds, the profile it presents makes it a bit akin to pushing a brick wall through the air at high speed though that's little different than other cruisers.
The RIII handles surprisingly well but that's in large part due to its relatively low center of gravity. Lean angle is limited by the floorboards on the Classic and Touring models, mine have significant metal ground off the outside lower edge. The situation is a bit better on the standard RIII since it lacks floorboards but even there a slightly above average rider should have no trouble grinding the hero blobs off the pegs in fairly short order. The bike itself is more capable than its design allows. However that's the case with pretty much all low-slung cruisers ... which is to say that's pretty much the case with cruisers in general. I've yet to ride a cruiser whose cornering ability wasn't limited by foot pegs or some other bit that touched down before, usually well before, the limits of the bike's traction.
The idea that an RIII is going to suffer in the twisties compared to other cruisers is just silly. If anything it has an advantage because it's far easier and much quicker launching on exits than most anything else on the road. Riders who keep the RIII in the mid-high RPM range will make up on corner exits much of what they lose to their sport-touring companions (and surpass most cruiser riders) in pure cornering ability. There's a certain fluidity that comes from not having to dance on the gearshift through the twisties, that's purely a matter of torque on tap.
Weight might be an issue for some but the idea that the bike will flop uncontrollably on its side if a rider dares take it, "... about one degree off from vertical," is pure nonsense. Logically that can't be remotely true because if it were no one would be able to pick it up off its side stand, said stand supporting the Rocket at significantly more than one degree off vertical. When I purchased mine I ran into my sister (not literally) on her 600cc Shadow. She didn't even want to sit on the Rocket as she found its size intimidating ... err, bad pun not intended. But with a bit of coaxing I did manage to get her on the RIII at which time she picked it up off the side stand, rocked it gently from side-to-side a bit, grinned, and rode away on it (all 5'3" and 115# of her). When she (eventually) returned, with a huge stupid grin on her face, she commented that she found the bike very rideable but that she wouldn't want to wrestle it around every day in stop-and-go traffic. For her I think that's a fair comment but for average-sized and larger men I wouldn't think this would be an issue.
Where the weight might be an issue for some is in the twisties. If you're particularly small or weak you'll have trouble or tire out quickly from flicking the bike rapidly back and forth. It isn't a challenge but it is the one place where under normal riding conditions the weight of the bike is noticeable. Over a couple of days last spring I rode mine down the length of Skyline Drive and the Blue Ridge Parkway then did a half-dozen passes through Deal's Gap and headed home via the Cherohala Skyway; no fatigue here I'm something of a gym rat and bench a small car on good days. It's not a canyon carver and was never intended to be one. If that's what you're after pick up a middleweight or smaller sport bike and go at it. Right tool for the job and all that sort of thing y'know.
Mechanically mine's been flawless though with the rather amazing scope of the engineering project Triumph undertook in building this monstrosity I always make it a point to ask all other Rocket Captains that I run across about their experiences. Of course this is merely anecdotal evidence from a very limited subset including myself and a few other folks however I've yet to personally meet anyone who had a significant issue with an RIII. The worst identifiable pattern of mechanical failures out there is a cam chain gear on the earliest models that sometimes comes apart with catastrophic results for the engine. The moral of that particular story: don't buy an '04. Personally I'd skip the '05 models as well though the known shattering cam gear fault was corrected by then. FWIW the dealers I've spoken to (one BMW / Triumph and one Suzuki / Triumph / Victory) generally think that the Rockets are, if anything, slightly more reliable than average for the bikes leaving their shops. Heck, if I were worried about mechanical issues I'd never have bought that BMW GSA.
The RIII does run a bit warm. You will notice a bit of heat from the engine, particularly on warm days. The fan will run most of the time below 30 MPH on days above 70 degress or so. Personally I don't find the amount of heat bothersome or this behavior troubling but YMMV.
IMHO the Touring edition of the RIII is an abomination. The standard & Classic models feature a claimed 147HP & 143 ft-lbs of torque (which pretty much blows an aircraft-carrier-sized hole in the idea that the RIII is in any way, shape, or form "underpowered" as claimed above) but why Triumph would give up around 40 HP for very modest torque gains on the Touring model is beyond me. If I were speculating I'd guess it's because they put significantly narrower rubber on the Touring model (also a bad idea IMHO) and are worried about rear wheel slip. If you're considering an RIII Touring model, buy a Classic and slap on the Corbin Beetle Bags. You'll then have a better bike than the RIII Touring and will be lacking only that oh-so-vintage-looking tank speedo; a fair trade for a Sportster's worth of extra HP IMHO.
The neutering through the first 3 gears (~10% reduction on power via restrictions on the opening of the secondary butterflies) came about prior to RIII production when test riders commented on wheel slip. A couple of my sport bike riding friends, upon returning from rides on my bone stock RIII Classic, chastised me for failing to inform them that whacking the throttle really hard would indeed point the front wheel skyward and that it was entirely possible to break the back wheel loose on 1st/2nd and 2nd/3rd shifts on clean dry pavement if the rider was a bit aggressive with the bike. They very much enjoyed those sorts of hooligan antics but would have appreciated my telling them in advance that the RIII will act much like a sport bike if it is ridden like one. If for some reason actual measured RWHP in the low 130s (which is about where the claimed 147 HP models run on a dyno) is insufficient more is easily had. Drop the catalytic converter for a bypass pipe, install an underseat K&N filter, and use Tuneboy: to correct the factory-lean fuel / air mix, advance the ignition slightly, and remove the restrictions on the secondaries and you’ll be running high 140s on the dyno with more available if you want to hassle with a new exhaust. For the "too much is not enough" crowd there's a $6k bolt-on turbo that will bring the RWHP & torque both up over 200. Probably worth noting that a bone-stock RIII already has more HP & torque than a 96 cu-in HD with a similar turbo attached.
Personally I don't think that the RIII would be a good replacement for an LT. I'm not nearly cushy or pampered enough to need something like an LT or a 'wing but if I were it would be tough to go back to an unfaired bare bones rig like the Rocket. I did a 5k trip on mine with only Corbin bags and found that it worked just fine but then I don't mind rain, cold, buffeting, lack of GPS / stereo / intercoms /cupholders/etc. You LT riders purchased LTs for a reason, I doubt very much that you'd be happy with an RIII as a replacement. Triumph is still a fairly small marquee and the RIII is still a niche product so don’t buy one believing that you can outfit it like a full-on touring rig, the equipment just isn’t available even if you wanted to do it. Be very sure that you can get what you need to kit it out before you proceed.
So why buy an RIII at all? For me personally motorcycles succeed if they're a joy to ride. The RIII is that and more, it really is great fun and, unlike a lot of bikes I've owned, remains great fun a couple of years and many thousands of miles into the ownership experience. It's outrageous, eccentric and cartoonish in its proportions. It takes something of that sort of rider to enjoy one I think. Personally I hope none of you buy one as I do so hate to meet myself riding 'round the block.