A guardians favor healadin geared to flag carry can make a mean flag carrier. I play this style and I can say though it has certain faults (mainly purging my freedom and not being able to use my major CD's while holding the flag) it also has great benefits if you can get the tricks down. Druid flag carrying is always going to be looked at as better, personally i think it also has its own faults and shines more for bringing the flag over through mid. When you're caught indoors to cap the flag a pally does better than a druid. I have to rely on proper positioning, heals and gear like gnomish & swift boots to make a flag play through mid field. I do come with Loh though and on a 40min CD with a 30% armor increase for 2mins after use this is almost always enough to tank the entire enemy team without going down.
Having a mage flag carry for the reason that they can blink nova and iceblock re-grab the flag usually ends up with your mage being one/two shot be the enemy nukers, again its great for getting through mid but with a big enemy offence looming nearby that mage can easily just peel for a built flag carrier. Also note worthy is you can bubble/cancel to re-grab the flag just like the iceblock trick.
A prot warrior flag carrier is a good arguement as second best but is more reliant on his own team than a healadin because, the lack of self healing & no way of removing snares and movement impairing effects. While I have BoF on a 20 second CD that lasts 16 seconds wich equates to 80% of the time you can't stop me from running wich I can then also amplify with one of my many speed boosts ( swfitness/gnomish/goblin rocket boots & swiftness potions). I feel like having no way to actually heal yourself is the biggest downfall wich makes it much better for premades rather than solo Q heroes.
Viper sting isn't ever a big deal to me just be quick with your self dispells, hunters go oom trying to spam me with it. It really only hurts when you're being heavily pressured and need to spam heal yourself to stay alive, not even being able to get a gcd in to hault the mana loss.
Everything is situational of course, but let me tell you anytime i've let one of these druids Fc over me they somehow drop like their wearing cloth. I don;t even wear a full mail set since i use arachnid gloves of the eagle and green lens of healing but I seem to take less damage. Having only 137+ healing my heals are pretty decent and my mana pool is sizable. If i'm tanking a couple people and juke the counter or kick i can spam flash of light on myself wich can keep me alive with around 2-4 people depending on how much damage they can output. Comparatively a druid being forced to self heal needs to shape shift out of bear form and quickly heal & shape shift back to not be caught in his squishy form. This doesn't matter as much when you already have other healers with you, wich makes the point of a druid would be better to take an offensive head on while you let the pally heal instead, since they can bubble/bop themselves while keeping the druid up. So be it, I still always ask to flag run over druids unless I feel like the enemy team will be likely to stop me from crossing mid.
Overall I think you're underrating paladin Fc a bit. I'm not sure wether for lack of knowledge or from what you have personally seen in game but, i'm just throwing my 2 cents in there since I specifically made an paladin to be a flag runner and have some experience to share.
I'd say that in premades, a prot Warrior is probably best, since it can actually fight and never go oom and if it has pocket heals all day it can survive whatever. I don't think there's much question about the theorycraft of a prot Warrior being best in premades.
I'd like to talk about my experiences in pugs. I've run as a Druid, Mage, and Paladin FC in 39s before, though it was outside of classic and during BC and later expansions. All three had their own unique advantages and disadvantages and all 3 were fun to play.
I think that the first thing is that player skill trumps whatever class you're running. The FC position in a pug requires extensive knowledge of the game and positioning is the most important in the FC role. This skill at positioning is vastly more important to anything else, such as class choice.
Now let's go into the three classes I've played and what I think their advantages and disadvantages are.
Arcane Mage flag carrying was by far the most fun and rewarding gameplay I've ever experienced playing WoW. In later expansions and beyond, it was in my opinion the most broken pug flag carrier. I'm currently playing one in classic where a few of the arcane talents aren't as good and mana shield does not block magic damage so I'm giving the nod to Druids. The ability to blink away from stuns, polymorph, resist spells, CC, AoE, presence of mind, silence casters for 4 seconds, and chill melee attackers makes mage a lot of fun and versatile to play.
The main disadvantage to an FC Mage is the inability to deal with getting overwhelmed. Getting stunned, slowed, CC'd, etc and then rammed by multiple enemies usually results in death. Having multiple enemies on top of a Mage usually means the Mage has to spam mana shield and then goes OOM quickly and then dies. Mages are especially vulnerable to slows and one of my top priorities while playing a Mage is trying to avoid them.
Druids are definitely the best pug FC class in classic WoW. Their movement is unrivaled. Without mounts in classic 39, this unrivaled movement is very useful. In my opinion, when you say that handing the flag to a Druid results in them dying quickly, it's due to their player skill. I think that in the right hands, a Druid is the best pug flag carrier in classic. Like you mention, it's exceptional at getting across mid. On top of that, it has a virtual invulnerability to slows, a tanky form, superior movement speed even indoors (with talented cat form), instant 30 yard cast moonfire to slow hunters, fairy fire against rogues, self-healing, and entangling roots/nature's grasp.
The main disadvantage to a Druid FC is inability to deal with stuns. When I played a Druid, I was consistently frustrated with playing against Rogues. Getting cheap shotted/kidney shotted and being out of bear form (or even in bear form!) was really difficult to deal with, and then you add in blind and crip poison and vanishing out of roots. I'm way happier playing a Mage because I can actually deal with Rogues. Also, I think a Druid's self-healing can be mediocre and slow sometimes. A Druid's ability to self-heal basically relies on the Druid maintaining distance on others and not getting overwhelmed.
Paladin flag carrying is extraordinarily versatile. I think the best thing a Paladin has compared to the other two classes mentioned earlier is its ability to self-heal. Paladins are tanky with mail armor and heal for a good amount. Like you said, they can remove viper sting and have BoF. In addition, Paladins are okay brawling against several opponents and it takes forever to get a Paladin to go OOM. Divine Shield and Concentration Aura are super useful tools. Rogues can be dealt with through tankiness, healing, and fake casting.
The main disadvantage to a Paladin is the lack of movement advantage. It's difficult to gain distance on other classes. With just hammer of justice as a gap-distancing tool, playing a Paladin FC in classic seems like it relies on its ability to self-heal and tank. In later expansions, Paladins gained a judgement that restricted a target's movement to 100% and that's the kind of gap tool I relied on during my time playing as a Paladin FC. It seems difficult playing without it. It seems like you're confined to self-healing and praying (literally, with Lay on Hands) that your defense can eventually kill the opponent's offense.