All the usual features of an Everyman puzzle in this one.
Abbreviations
cd cryptic definition
dd double definition
cad clue as definition
(xxxx)* anagram
anagrind = anagram indicator
[x] letter(s) removed
definitions are underlined
Across
1 Chorister‘s ablution with regular purging
ALTO
The odd letters of AbLuTiOn.
4 Last two pieces missing from puzzle depicting lively dances
JIGS
JIGS[AW]
8 Cause aggravation – by rustling?
GET SOMEONE’S GOAT
A cd. ‘Rustling’ here needs to be understood in the meaning of stealing animals. The phrase is originally AmEng, for those who get exercised about such things.
11 Drink in a deceit, haplessly
ICED TEA
(A DECEIT)* with ‘haplessly’ as the anagrind.
12 Copper to steal from the French museum’s entrance? A concern
PROBLEM
A charade of P, ROB, LE and M for the initial letter of ‘museum’. P is for ‘penny’ or ‘copper’ coin.
13 Extra time, erroneously, in cabbie’s device
TAXI METER
(EXTRA TIME)* with ‘erroneously’ as the anagrind. This is a great and succinct anagram, but I can’t remember seeing it before.
14 Passed quickly – for feedback?
RAN BY
A dd. For the second definition, you have to take the whole clue as the definition, I think.
15 Doctor who may be a smoker
CURER
A much clearer dd.
16 Slap-happy greetings?
HIGH-FIVES
A cd.
19 Something tells you I’m not a horse, and that’s the point
UNICORN
Another cd. A UNICORN is a horse with a point.
21 Sample taken from cover (duet), Not Before Time
OVERDUE
Hidden in cOVER DUEt.
22 Thirtieth: then go bananas or behave appropriately?
HIT THE RIGHT NOTE
(THIRTIETH THEN GO)* with ‘bananas’ as the anagrind.
23 Most important thing is vintage cross
CRUX
A charade of CRU and X. CRU is ultimately derived the French verb croître, meaning ‘to grow’ and is used in expressions like Premier Cru.
24 A Freudian concept, to begin with, scarcely helps
AIDS
A charade of A, ID and S for the initial letter of ‘scarcely’.
Down
2 Chairperson suppressing pride?
LION TAMER
A cd. The classic cartoon image of the LION TAMER is with a whip and a chair. And lions come in prides.
3 Cavort around, ‘entertaining’ Everyman’s beginning to ham it up
OVERACT
An insertion of E for the initial letter of ‘Everyman’ in (CAVORT)* The anagrind is ‘around’ and the insertion indicator is ‘entertaining’.
4 Judge unwise: unripe berry used in booze
JUNIPER
A charade of J and (UNRIPE)* The anagrind is ‘unwise’.
5 Blast! Zero enthusiasm
GUSTO
A charade of GUST and O.
6 Assuming ‘geriatrics’ enfeebled, I’m seldom tolerant, primarily?
AGEIST
The initial letters of the first six words of the clue and a cad.
7 Smiley perhaps pulling in gut, short and stout
STUMPY
An insertion of TUM in SPY. The insertion indicator is ‘pulling in’. [George] Smiley was a secret agent in the novels of John Le Carré.
9 Heroic text’s adapted as scary film
THE EXORCIST
(HEROIC TEXTS)* with ‘adapted’ as the anagrind.
10 Funny one I loved in YouTube clip?
ONLINE VIDEO
(ONE I LOVED IN)* with ‘funny’ as the anagrind.
14 Pondered what lake’s surface may have done
REFLECTED
A dd.
15 Apprehended and, where hearings take place, heard
CAUGHT
A soundalike clue (‘heard’) for COURT.
16 Voodoo Child rocker Jimi ultimately bearing number that’s entertained?
HENDRIX
Given the crossers and the definition, the answer was never in doubt; but I struggled to parse this for the longest time. Since it’s a down clue, I think it is this: DRI for the final letters of ‘Child’, ‘rocker’ and ‘Jimi’, ‘bearing’ N for number, giving us NDRI, and all that inserted into HEX. HEX is a spell, so the insertion indicator is ‘in a spell’, in other words in HEX. And if someone is ‘in a spell’ or ‘spellbound’ then they are ‘entertained’. Others may have a different interpretation, including where the definition ends.
17 Artificial intelligence, mostly Grok for example, rising in former Soviet state
GEORGIA
A reversal (‘rising’, since it’s a down clue) of AI, GRO[K] and EG.
18 Appeared to be what you can do on Riviera
SEEMED
If you are on the Côte d’Azur, you can certainly SEE [the] MED.
20 To the rear, somewhat different
OTHER
Hidden in tO THE Rear.
Many thanks to Everyman as always for this Sunday’s puzzle.

I had also failed to come up with a satisfactory parsing for 16d, although as Pierre says the answer was always obvious. Pierre’s solution has inspired me though; I think it’s Voodoo= HEX with NDRI as per his solution inserted (entertained).
Thanks Pierre & Everyman
22a. HIT THE RIGHT TONE would be an equally plausible answer based on the anagram, (it was one I certainly considered); however, knowledge of Everyman’s penchant for a symmetrically placed rhyming pair comes to our assistance and the conclusion that the intended answer must be NOTE to rhyme with GOAT in 8a !
Thanks for the blog , I think Graham@1 has the final version for HENDRIX , not the best clue . The original track was actually Voodoo Chile , a shorter version based on this was called Voodoo Chil[e/d] ( slight return ) , depending on the record label . The shorter version later a UK single and now just Voodoo Chile . Complicated , but Jimi clearly sings Chile .