I think you're bypassing the "don" and "doff" rule for shields.

I agree that there's daylight to be seen between carrying and "wielding" a shield. But I'll pose you this question: if you can have a shield and switch states from "wearing" to "wielding" it freely, then what is donning or doffing?

As for the comparison to equipping a weapon, I don't think it's valid. No weapon classes are described as requiring an equip-time; all armor are described as needing time to make usable. (I'm ignoring the little bit of time-constraint provided by "loading".)

My read of your description above is that you've house-ruled the donning and doffing of a shield to cost zero in the action economy. If that works for your group, that's great. But I think it is a departure from PHB rules.

For mundane weapons, you have already found the answer.

• @Eternallord66 points out that warhammers would be a "best" choice, or flails; there are 12 monsters vulnerable to bludgeoning damage, 0 vulnerable to slashing damage, and 0 vulnerable to mundane piercing damage.

For magical weapons: a moonblade that has multiple runes on it and a Rod of Lordly Might.

Since the history and runic additions for a given moonblade are completely at the mercy of the DM who has that weapon in their world, two +3 warhammers/longswords/rapiers/battlezses would seem to be a more "practical" max since attunement to the weapons is not required.

But, if you have access to magical items that are "legendary" in your campaign ...

A more "practical" answer: a Rod of Lordly Might and a +3 weapon

If you push button two, you'd get two +3 weapons.
If you push botton one, you'd get a flame tongues which each do 2d6 additional damage, and your +3 weapon. The flame tongue adds seven damage on average unless the target is fire immune/resistant.

The above does not rely on the "how many previous owners" problem that the moonblade answer does.

Could you use a Flame Tongue and a Rod of Lordly Might together?

Technically yes. You can push button two to get a second flame tongue from the Rod, and end up with two flame tongues. Even though you are attuned to what looks like two flame tongues, you are actually attuned to a flame tongue and a rod of Lordly Might.

If you can find them, you want a Rod of Lordly Might and a Flame Tongue

(Longsword or rapier for the flame tongue so that it's one handed)

Thanks to @V2Blast for reminding me that you can't attune to two copies of the same weapon