PVE Protection Paladin Tank Gear & Best In Slot (BiS)

Blade of the Archmage:

• 1.8 Weapon Speed (Over the course of 1 minute, that would be about 34 attacks – 60 ÷ 1.8 = 33 + your initial melee attack opener)
• Seal of Righteousness causes 100 damage per successful melee attack (269 spell damage)
• Assuming no attacks miss, are dodged, parried, or blocked, it would be 34 x 100 = 3,400 (3,400 x 1.90 = 6,460 threat or 107.67 tps)

 
Gavel of Unearthed Secrets:

• 2.70 Weapon Speed (Over the course of 1 minute, that would be about 23 attacks – 60 ÷ 2.70 = 22 + your initial melee attack opener)
• Seal of Righteousness causes 150 damage per successful melee attack (269 spell damage)
• Assuming no attacks miss, are dodged, parried, or blocked, it would be 23 x 150 = 3,450 (3,450 x 1.90 = 6,555 threat or 109.25 tps)

Seal of Righteousness’ DoT effect cannot be critical strikes, so Blade of the Archmage’s 0.21% additional critical strike rating does not come into play.

Applying the Seal of Vengeance DoT is just a chance-on-hit. It was not guaranteed to happen on each melee attack until Patch 3.0.2, which saw some other changes to it, too. If you look at or conduct some tests, you will see that there are times where each weapon will generate those stacks faster than the other – this is because it is a chance to apply a stack, and not guaranteed to happen.
 
You are correct in terms of saying that with Blade of the Archmage we would have a higher likelihood to receive more procs from Seal of Light or Seal of Wisdom (Assuming you also meant the Judgement application, as well) The question is whether the increase in either resource’s regeneration is worth it.

Using the above-mentioned statistics and timeframe (assuming no misses or dodges), we have the potential to see:
 
Seal of Light:

• Gavel of Unearthed Secrets: 3,059 Healing
• 50.98 HPS
• Blade of the Archmage: 4,522 Healing
• 75.37 HPS

Judgement of Light:

• Gavel of Unearthed Secrets: 2,185 Healing
• 36.42 HPS
• Blade of the Archmage: 3,230 Healing
• 53.83 HPS

Seal of Wisdom:

• Gavel of Unearthed Secrets: 2,783 Mana
• 46.38 MPS
• Blade of the Archmage: 4,114 Mana
• 68.57 MPS

Judgement of Wisdom:

• Gavel of Unearthed Secrets: 1,702 Mana
• 28.37 MPS
• Blade of the Archmage: 2,516 Mana
• 41.93 MPS

You are correct in saying that Continuum Blade DOES have more stamina (On a Protection Paladin, that 6 becomes 7 – so, our health increases by 70) We’ve already deduced that faster weapons don’t necessarily cause more damage or threat with Seal of Righteousness, and applying Seal of Vengeance stacks is not guaranteed to be faster. Because of this, Continuum Blade would only appeal to Paladins for the +70 Health and +0.63% Spell Hit Chance. With Continuum Blade equipped, our spell damage is 231:

• Seal of Righteousness: 100.13 TPS
• You’re losing TPS potential using this weapon while only gaining 70 Health and 0.63% Spell Hit Chance. It’s not worth it in this case.

Sacrificing 38 spell damage (ignoring the critical strike rating at this point) is like sacrificing 9.03 TPS from Consecration – an ability that does not rely on Spell Hit Rating. No one is denying that spell hit rating is beneficial, I’m saying that given how our spell damage affects our Consecration damage, it’s more beneficial to go for +9.03 TPS than it would be to get 0.63% Spell Hit given the fact that it’s RNG – we don’t know how many times Exorcism will miss, or any ability for that matter. Consecration is guaranteed to hit 8 times per rotation.

“There are times where gavel will reach 5 stacks at the same rate or faster than the swords BUT the swords will do it faster more often.”

That’s not necessarily true. If we both flipped a coin, and you flipped faster than I did, hoping for 5 heads, I could still beat you every time to 5 tails. The same rule applies for Vengeance stacking. Yes, given the speed, you have more opportunity for it to stack, but that doesn’t mean it WILL stack any faster or AS fast as with the Gavel. Will it sometimes? Absolutely. Will it not sometimes? Absolutely. That’s the beauty of RNG – it’s not something we can know when something will proc or be triggered, however you’d like to say it. I gather you’ve run some tests, what were your average results?

We can agree to disagree. There’s nothing wrong with that.

I’ve never heard of, nor experienced myself, Seal of Vengeance stacks taking more than a minute to reach 5. That sounds like an extremely long run of bad luck.