Yes it does say that for shield, but not for Dex.
It says: 'The armor (and shield) you wear determines your base Armor Class.' That's pretty unambiguous. Although it doesn't repeat that level of clarity under the heading of Shield itself.
That's because Shields can be used with Mage Armor. When wearing armor, they are part of the base AC. When not wearing armor, they are a modifier. The same seems to be the RAI implication of Dex.
It also says 'you add your Dexterity modifier to the base number from your armor type to determine your Armor Class' (not to determine your base Armor Class).
On page 177 is also specifies that 'you might add some or all of your Dexterity modifier to your Armor Class', not base Armor Class.
Regardless, the example of Druid and Dryad indicates that Barkskin does not stack with Dex. Minimally, Dex is not a modifier to Barkskin. This seems to indicate that it is possible that a Shield also is not a modifier to Barkskin.
I do agree with you that Dex is not spelled out as part of base AC when wearing armor. But, it seems to be RAI, even if it is not RAW.
For example, base AC for Mage Armor is 3 + Dex modifier. If base AC did not include the Dex modifier, than there would be an argument (not very strong) that spell casters could have an AC of 13 + Dex (i.e. base AC + Dex when using Mage Armor.
Hence, the implication that Dex is part of base AC from the chart on page 145 is a reasonable one. The shield modifier on that chart is part of base AC. The Dex modifier might change based on the armor worn, but base AC always changes based on armor worn.
Definitely not RAW, just possibly RAI.
It also sounds a bit counterintuitive that something that is part of your base AC can vary in application, rather than a modifier. Your modified AC can be situational, but your base AC shouldn't change. The fact that your Dexterity modifier functions differently with different types of armor means (to me) that it's a modifier, not base.
Your base AC changes whenever your PC situationally puts on different armor. Or your PC takes it off to go to bed. Or when Mage Armor is cast on your PC.
This is also consistent with earlier editions where certain circumstances denied your Dexterity modifier to AC (flat-footed, etc.).
But earlier editions do not matter.
I think it is totally reasonable for the minimum AC 16 of Barkskin to be considered to be base armor and normal base armor includes armor, Dex, and Shield from a RAI POV.
If Dex and Shield are just added to the 16 of Barkskin, that can be AC 23 at level 8. That doesn't seem like it is RAI of the Barkskin spell with the "can't be less than 16" wording. Barkskin seems to be a spell intended for when Druids wild shape, or for Ranger companions (as a general rule). Obviously it can be used for more than that. But, being a spell to boost AC of the Ranger or multiclassed Rogue to 23 doesn't seem to be the intent.