Macbeth dates back to around 1606.
Tongue of Dog
While the most common name for the plant seems to have been hounds tongue, there are a few instances of it going by another name: Dogs tongue.
I found “Dogs toong” in the The Herball Or Generall Historie of Plantes (1597), page 337.
Here’s another quote, from the ME translation of Guy de Chauliac's Grande Chirurgie (?a1425):
Lefes of lingue canis i. dog tonge
However, the phrase also had its literal meaning at the time, so it could still be literal — or perhaps a reference to both meanings.
The unreferenced sources don't even tend to claim this is a plant. Sting means fang. It is most likely that blindworm refers to the slow worm (which is neither slow nor a worm nor blind nor deaf nor a snake: it is a lizard without limbs). Otherwise, blindworm sometimes refers to a kind of water snake or serpent. Or maybe a venomous snake.
Blind-worms and newts: Shakespeare's other works
The language of the witches is similar to that in Timon of Athens:
Engenders the black toad and adder blue,
The gilded newt and eyeless venom'd worm,
And yet again in A Midsummer Night's Dream:
You spotted snakes with double tongue,
Thorny hedgehogs, be not seen;
Newts and blind-worms, do no wrong,
Come not near our fairy queen.
These are the only occurrences of "newt" in Shakespeare's works. Every time the word appears, it's paired with the blind-worm (which also isn't poisonous, despite the popular belief). And in all the other sources there is no connection to any plant (or witches or herbalism) that would make sense.
They are poisonous plants, no matter how you look at it. The dark, nighttime digging of the root was said to make it more poisonous. Poison is a common theme in many of the ingredients.
Even after searching some, the closest I got to a connection between plants and the other ingredients is that the name “adders tongue” is also attested from around the same time period (eg in Historie). There really are a lot of plants that are referred to as tongues.
I don’t expect that you would be able to find even a hint of an alternative explanation for some of the ingredients being plants. For example, I don’t see another possibility for “witch’s mummy”: they’re calling for a bit of embalmed flesh or fluid. I don’t know about witches, but “mummy” was a common enough ingredient in medicine in the past. Taking again from Chirurgie:
Mummia is flesh of dede men balsamate [embalmed]
In fact, Johnson (of dictionary fame) lauds the scene for how gruesome many of the ingredients are:
It has been already mentioned in the law against witches, that they are supposed to take up dead bodies to use in enchantments, which was confessed by the woman whom King James examined, and who had of a dead body that was divided in one of their assemblies, two fingers for her share. It is observable that Shakespeare, on this great occasion, which involves the fate of a king, multiplies all the circumstances of horror. The babe, whose finger is used, must be strangled in its birth; the grease must not only be human, but must have dropped from a gibbet, the gibbet of a murderer; and even the sow, whose blood is used, must have offended nature by devouring her own farrow. These are touches of judgment and genius.
Origin and explanation of the claim?
The originator of this claim may have been Wiccan author Scott Cunningham. I can't access the book where it likely first appeared (Cunningham’s Encyclopedia of Magical Herbs, 1985), but he definitely did claim that every one of those ingredients in the brew is a plant. I can access most of his list of what he calls 'tentative "translations"' in Google Books (and what seems to be the entire list is on Quizlet).
The best explanation for the claim can be found in More than “toil and trouble”: Macbeth and medicine (the list of suggested alternatives is slightly different). The reasoning for some is that one word matches between the phrase used by the witches and an attested form of some plant (e.g. "scale of dragon" becomes dragon's blood). In some cases, the author doesn't even get this close to a tentative connection, and makes an association because they feel the plant resembles what the witches said (e.g. mustard seeds being "eyes of newt"). And for the rest, there's no connection that I can even guess.
These methods, of course, ignore specific wording when it is given, such as the fact the "finger" comes from a babe whose entire (albeit short) life is described in the recipe (the website's suggestion is that this is either Bloodroot or Cinquefoil, while Cunningham says it's "roots from a young dead tree"). By doing so, this eliminates all possible wordings for anything you could brew that is meant to be taken literally as not a plant. How such a person would explain away the other historical context (such as mummies) is beyond me.
What I do agree with: "Drawing any conclusion from a list like this requires many layers of assumption and speculation".