An Ancient Cryptic Puzzle

Question

Here's a variety cryptic for your puzzling pleasure. The clues are cryptic, but as you can see below, the grid is unusual. Axial clues are entered from the outside in. Around clues are entered in order according to a pattern created by the Josephus Problem. The first letter is entered in some location (not necessarily where the letter indicator for the ring is), and the subsequent letters are entered by skipping some fixed number of empty spaces, circling around the ring as necessary. The number of spaces skipped is fixed within each ring, but may vary between rings; note that none of the skip numbers is greater than 8. It is up to you to determine the initial location and the number of spaces skipped, but the clues are entered clockwise. I wish you luck!

Axial

1. One out of many! (3)
2. Alternative selection from bassist lays down bottom with a groove (3)
3. They deliver Downs' partner (3)
4. Almost entirely Canadian line drafted from front ranks of Red Wings, Oilers and Jets positions (3)
5. South's short for partner in clubs, requiring bid (3)
6. It's slippery around frozen precipitation after scraping perimeter (3)
7. Competition eliminating final, natural source of sugar (3)
8. It's one way star ejects corona (3)
9. Soldier's medical tests (3)
10. Doc prepping package for embedded cyst extraction (3)
11. Chargers' guard, center pushing between Dolphins' ends (3)
12. Starting fire late to identify place for ashes (3)
13. Teen? Not quite! (3)
14. Apply for job after wasting time (3)
15. Wrongly claim extraterritorial possession (3)
16. Dude goes about war...casually? (3)

Around the Rings
A1: Short-form writers are uncommonly terse after discarding some characters out of twenty (8)
A2: Short story of small girl, with brother, riding horse uphill (8)

(Original had these clues reversed in error, thanks to @jafe for catching.)
B1: Former king hangs around ample, toned model (8)
B2: Positioned top-knot after weaving sinew in creating vintage bag? (8)

C1: Equivocating government leaders, worry with street-walking ultimately force last of residents to find ways out (8)
C2: Airline, incorporating group of lawyers after I left, becomes no better (8)

D1: Putting laundered baht in action is final stop in lifecycle (8)
D2: All the People: variety show about mass movement (8)

Answer 1

Axial

1. One out of many! (3)
(-m)ANY &lit.

2. Alternative selection from bassist lays down bottom with a groove (3)
_A_S_S_

3. They deliver Downs' partner (3)
UPS ddef

4. Almost entirely Canadian line drafted from front ranks of Red Wings, Oilers and Jets positions (3)
D_+E_+W_ – Distant Early Warning Line, formed from the first letters of Detroit, Edmonton and Winnipeg

5. South's short for partner in clubs, requiring bid (3)
S+IS – Club requiring bid is a sorority, whose members are sisters (thanks OP)

6. It's slippery around frozen precipitation after scraping perimeter (3)
(-t)EEL(-s)<

7. Competition eliminating final, natural source of sugar (3)
BEE(-t)

8. It's one way star ejects corona (3)
(-d)ENE(-b) – East-northeast (which is "one way"), from the star name Deneb (thanks, Guest)

9. Soldier's medical tests (3)
G.I.+'S – Abbreviation for gastrointestinal tests

10. Doc prepping package for embedded cyst extraction (3)
_T EX_

11. Chargers' guard, center pushing between Dolphins' ends (3)
D_(_A_)S – Central letter of "guard" between first and last letter of "dolphins"; District Attorneys charge people with crimes.

12. Starting fire late to identify place for ashes (3)
(-b)URN

13. Teen? Not quite! (3)
TE(-e)N &lit.

14. Apply for job after wasting time (3)
(-t)ASK

15. Wrongly claim extraterritorial possession (3)
_ERR_

16. Dude goes about war...casually? (3)
NAM<

Around the rings

A1: Short-form writers are uncommonly terse after discarding some characters out of twenty (8)
TWE_+ETERS*

A2: Short story of small girl, with brother, riding horse uphill (8)
S+(YNOP+SIS)<

B1: Former king hangs around ample, toned model (8)
EX+EMPLA*+R

B2: Positioned top-knot after weaving sinew in creating vintage bag? (8)
WINES*+K_+IN

C1: Equivocating government leaders, worry with street-walking ultimately force last of residents to find ways out (8)
E_+G_+(-st)RESS+_E+_S

C2: Airline, incorporating group of lawyers after I left, becomes no better (8)
UN(-i)(ABA)TED – ABA is the American Bar Association

D1: Putting laundered baht in action is final stop in lifecycle (8)
DE(ATHB*)ED

D2: All the People: variety show about mass movement (8)
MAN+EUVER<

Figuring out the desired order for rings

Ring A fits the grid if we skip five spaces each time.



Ring B fits the grid if we skip two spaces each time.



As it happens, ring C also works if we skip two.



Skipping six works in ring D.

The finished grid

Feedback

This was a really enjoyable puzzle. I found it pretty difficult! A large part of the axial clues need to be filled without any help from the grid, since you can't rely on any letters from the around clues until many of the axial clues have been placed.

I ended up just trying all combinations to fit the around clues into the grid. Maybe there's a more clever way to work them in? It's not a pattern I can visualize in my head. Ring A was especially difficult, since neither of the first two letters were included in the checked squares, so there were a lot of possible combinations to check.

I'd advocate caution when using two- and three-letter acronyms, simply because there is such a large number of them. I was not aware of GI as a medical abbreviation and had to look that up, and I spent some time searching why DAS means "chargers" before getting the pun. (Apparently there are a lot of things named DAS, which is surely true for many three-letter acronyms.) It's good that the wordplay in those clues is relatively straightforward, so there's not much danger of finding an alternative answer that fits the clue.

Answer 2

$8$ is probably

ENE from Deneb, primary star in the constellation of Cygnus