טהרות: מיכתב כתיבן לא נצרכא אלא לשיעור מקוה דלא כתיבא דתניא ורחץ (את בשרו) במים במי מקוה את כל בשרו מים שכל גופו עולה בהן וכמה הן אמה על אמה ברום שלש אמות ושיערו חכמים מי מקוה ארבעים סאה
- Talmud Tractate Chagigah 11a
Here the Talmud discusses the origin of Mikvaos and says (paraphrased translation mine):
"The concept of ritual cleansing is written in the Torah (Bible), but the amount of water is not spelled out. For we have learned: (quoting verse in Leviticus, 15:16) "..and he shall wash with water.." This means that he must bathe in a ritual bath (Mikvah). "...his entire flesh.." This means to teach that he must immerse in enough water that can cover his entire body. And how much is that? It is the amount of water contained in a space measuring an "Amah" by an "Amah", by 3 "Amos" high. And the Sages measured this to be 40 "Seah" of water.
(This is roughly 2 feet by 2 feet by 6 feet filled with water, (opinions vary) . Rav Yaakov Kanievsky (in his work: "Shiurin shel Torah") gives the modern amount at about 750 litres which is about 200 U.S. gallons of water. (again opinions vary))
As you can see, the Talmud understood the verse to mean immersion; and the verse itself suggests immersion because it says "his entire flesh", which is taken to be an act that can simultaneously cause water to wash the entire body at once.
This is also supported by the verse in leviticus 11:36:
"But a spring, or a cistern, a gathering of water (Mikveh), will be clean..."
This is a proof that the Torah grants ritual cleanliness only to a source of water attached to the ground. So it is immersion, and not sponging water all over the body limb by limb.
There is no Talmudic opinion that allows removal of ritual impurity of a Biblical nature to be cleansed by rubbing water over the body. Rather the entire Talmud is full of examples and laws about immersion.
So now that we know that the Talmud reflects an absolute opinion on Biblical Mikvah being an immersion into a body of water in the ground, what do we do about the OP's observation?:
"Archaeology has uncovered hundreds of ancient Mikvaot in Israel. They all date from the Second Temple era or later."
There are two answers:
1) First, I would not expect too many finds of stepped mikvah structures to be discovered from the times of Joshuah because people were not so into building lavish structures back then, and would have been more used to immersing in lakes, rivers, springs, cave-pools, or the ocean.
So what about the 1st Temple period?
The Ophel City Walls Archaeological site (also known as the "Water Gate") in the Jerusalem National Park area, had a new section opened to public view in 2011.
Although the site area has "layers" of discovery from Ummayad, Byzantine, Roman, Herodian and Hasmonean eras; it also has been revealed to expose in its deepest area, 1st Temple and "King Solomon" aged structures and pottery from the 8th - 10th centuries B.C.E.
And among the areas dated to 1st Temple times are to be found quite a few Mikvahs.
Kathleen Kenyon claimed the area was 8-10th century, and the current excavators/researchers under the Israeli Antiquities department, the late archaeologist Benjamin Mazar and his granddaughter Eilat Mazar, agree.
2) There is an "age old" trap that laymen tend to fall into when assuming things about the "science" of archaeology. They think its like math or developing new drugs. A man with a degree in archaeology should be just as trustworthy as someone with a dental license operating on a root canal?
NOPE. It just isn't true.
Archaeology has only been around for about 200 years. It has not proven to be an exact science. Major so-called authorities argue about the "facts" extremely and quite often.
Furthermore, most dig sites have been traditionally funded by royalty or government; specifically England and specifically in Egypt, since digs are expensive, and Egypt was a popular target of the rich.
Israeli archaeology is still in its infancy. New things are being found every day. It also takes many many years to advance in the field due to:
The speed at which you can gently sift and brush away layers of earth so you do not damage anything.
Finding the right place
Getting funded, and finding volunteers to help.
Dealing with war, and local upheaval
Government red tape
Being stopped for years on and off because the Arabs protest dig sites thinking you are going to damage Arab interests; as well as religious Jews (and others) worrying about violating graves.
Family politics - Archaeology is an inherited science where the community respects the rights of individuals and their close students or blood relatives, to claim exclusive access to materials and sites.
(That's one reason why the Dead Sea Scrolls found in 1947-49 have taken so long to be revealed to the public (30 - 60 years!) due to the refusal of some scholars to part with their scrolls in their possession.)
So there are two ways to argue for a claim:
1) based on finding evidence
2) based on not finding evidence. This is called the "argument from silence" and is usually considered a logical fallacy or a very weak argument.
The British press announced that archaeologists dug outside London and discovered a network of metal scrap wires dated to 500 BCE, proving that ancient Britons had invented the telegraph!
The French news announced that archaeologists dug outside Paris and found a network of small fiber particles dated to 600 BCE proving that the ancient Franks had fiber-optics!
The Israeli Arutz-Sheva news announced in return that archaeologists outside Tel-Aviv dug and dug and dug and dug and found ABSOLUTELY NOTHING! ... proving that ancient Israeites must have already invented cellphones!
So if you have not found any Mikvahs from before the 2nd Temple, it doesn't mean that it is "proof" they didn't exist; rather it means you need to buy a shovel and go dig. :)
I hope this helps. :)