Short Answer
No, your Tuckahoe species is likely Wolfiporia extensa.
Long Answer
Well, it sounds like there is some colloquial contradictions.
It could also be Peltandra virginica, which also has the common name tuckahoe.
This is an emergent perennial herb growing from a large rhizome and producing many large leaves. An individual leaf may have a petiole nearly a meter long and a blade half a meter in length. The leaves are quite variable in shape and size, but they are often generally arrowhead-shaped.
Kalm thinks the Tuckahoo of Carolina is the same plant...call[ed] (from the Indian name) Taw-ho, Taw-king, or Tuck-ah, -- namely the Virginia Wake-Robin, Arum virginianum [now Peltandra virginica]
However, it seems unlikely that this plant is the same as the "fungus-like" structure lacking leaves and stems that your source cites.
According to John Torrey's series of 1819 letters about Tuckahoe printed in "The Medical Repository of Original Essays and Intelligence..." (volume vi; Mitchill et al., 1891, pp 34-44)
This substance was first described by Clayton in his Flora Virginica: it is there called Lycoperdon Solidu...
Barton...mentions that the roots of the Smilax China [pseudo China] are used as food by the Southern Indians : but whether they are the same as the Tuckahoe, we can only conjecture.
And...
...it is generally supposed to be the root of the Convolvulus Ponduratus, called in this neighborhood the man of the earth; but... roots of the Convolvulus , which I have examined, were altogether unlike Tuckahoe.
...it is also believed by the natives in the South to be the roots of the Dioscorea villosa...
And finally...
[Based on its] arrangement the Tuckahoe must be considered as a Sclerotium...Our Sclerotium is so distinct from any described [previously] by Persoon, that I propose it as a new species, under the name of S. giganteum
So, given all that:
As I stated above, Peltandra virginica is unlikely the Tuckahoe you're interested in. Although it has a tuberous root, the common leaves and stems associated with the plant mostly preclude it from being correct.
I can't find any reference to Lycoperdon Solidu that doesn't date to the 1800s or talk strictly about Tuckahoe, so it's probably worth assuming that this name did not stick.
Smilax China is a climbing plant of the greenbrier genus. The leaves and stems of this plant would almost certainyl have been conspicuous, so this is also unlikely correct. The confusion likely lies in the "tuber" that develops in some smilax species (e.g., see below for a root of S. bona nox):
I also can't find any internet presence about Convolvulus ponduratus; however, we'll take Dr. John Torrey's word for it, and assume that this is not the proper species either.
Dioscorea villosa (or the wild yam) is a vine with many leaves, so it too seems to unlikely be the source of the Tuckahoe.
So...it seems like Torrey's writings deciding that this organism was a fungus and should be called Sclerotium giganteum is probably the best early source you have for this species.
Though sometimes found emerging from the earth and exposing a small part of the surface, it is generally met with 2 or 3 feet below the soil. When first dug up, it is soft enough to be easily cut with a knife, and of an acrid taste. Its colour internally is white, like that of the meat of the cocoa-nut, and its texture compact and homogeneous. It is covered with a tough substance, strongly adhering to the white parenchyma, of a dark brown colour, and somewhat wrinkled. When dried, the internal substance becomes hard and loses its acrimony, possessing very little taste or smell, and capable of being reduced to powder with out difficulty. When examined by the microscope, the tuckahoe exhibits no fibres or pores or any other indications of organization, so easily detected in roots and other vegetable productions of ordinary growth.
However, although Torrey claims to name the species (in 1819), you'll notice that the full name of the species you guessed in your question is Sclerotium giganteum Rostr., 1889, suggesting that the S. giganteum synonym of "Polyporus umbellatus (Pers.) Fr., 1821" was originally named 70 years after Torrey and therefore might in fact be a different species altogether....
In fact, the Tuckahoe of the American Southeast IS almost certainly a different species than Polyporus umbellatus, because P. umbellatus is a European species!!
Further, Wikipedia's description for Polyporus umbellatus (below) does not match those of early descriptions of Tuckahoe:
The fruiting body is composed of numerous (sometimes several hundred) caps. They are 1–4 cm in diameter, deeply umbilicate, light brown, and form the extremities of a strong, many branched stalk. The compound fungus can be up to 40 cm in diameter. The pores are narrow and white. The stalk is whitish grey, and originates from a strong, tuber like nodule that is underground. The flesh is white, rather soft when young, although hardens with age.
So let's turn to "Some Characteristics of the Southern Tuckahoe" by John A. Elliott (1922) in Mycologia, Volume 14, who refers to the tuckahoe as Pachyma cocos:
...suggested Pachyma cocos may... be connected with the genus Lentinus.
If we look up Pachyma cocos (i.e., "Pachyma cocos Fr. (1822)") it is now known as Wolfiporia extensa, and has the following description:
Wolfiporia extensa (Peck) Ginns (syn. Poria cocos F.A.Wolf) is a fungus in the family Polyporaceae. It is a wood-decay fungus but has a subterranean growth habit. It is notable in the development of a large, long-lasting underground sclerotium that resembles a small coconut. This sclerotium called "(Chinese) Tuckahoe" or fu-ling, is not the same as the true tuckahoe used as Indian bread by Native Americans, which is the arrow arum, Peltandra virginica, a flowering tuberous plant in the arum family. W. extensa is also used extensively as a medicinal mushroom in Chinese medicine.
Wait, Chinese??! I could have sworn we were on the right track!
Conclusion
Your Tuckahoe is likely the sclerotium that belongs to what is now known as Wolfiporia extensa.
