Note: This will be revised at least once a year. Probably the next revision will be in May 2023.
It is understood that TD Paine will not captain Australia again.
Updated basic statistics can be seen here: https://stats.espncricinfo.com/ci/content/records/283404.html
Q de Kock and Mohammad Rizwan became the latest of the relatively few wicket-keepers to have captained in a Test. Here is the full list (as on 20 Oct 2022):

Relatively few keepers have had extended runs as captains, although the first 3 were playing after 2000.
There are some like Gilchrist, Boucher and Jacobs who captained only when the regular captain was temporarily unavailable. Some, like DL Murray and IDS Smith, had long Test careers but captained their side in precisely one Test.
Sarfaraz Ahmed (13) was replaced after his last Test as captain in January 2019, and TD Paine (23) was replaced for non-cricketing reasons in late 2021.
Dhoni has the most dismissals, followed by Paine and Alexander at a long distance.
Dhoni has the most stumpings (24) and is followed by PM Sherwell with 16.
Lee Germon is a special case as he captained his side in all his 12 Tests. He is the only player from an established side to have captained in a Test on his debut in recent years. The last before him was Tony Lewis in 1972. (Obviously we do not count Sri Lanka, Zimbabwe, Bangladesh, Ireland, Afghanistan and the returning South Africa).
Germon is second on the list of those who captained their side in all their Tests.
The leader in this category was also a keeper:
Long ago PM Sherwell led South Africa in all his 13 Tests, a record for all countries. He was keeper in all these 13 Tests.
A few such as AJ Stewart played numerous Tests as keeper and non-keeper. The above list only includes the Tests where he was keeper. He also opened the batting in some of these matches. He captained England a total of 15 times, 12 as keeper and 3 as non-keeper. BB McCullum captained New Zealand 31 times, but was keeper in only 1 of them.
Other examples include KC Sangakkara (captained in 15 Tests, never as a keeper), AB de Villiers (captained in 3 Tests, none as keeper), R Dravid (25 Tests, none as keeper) and LD Chandimal (captained in 10 Tests, none as keeper). But they have captained in ODIs and T20Is as keepers.
Note that Taibu somehow took one catch as a fielder when he temporarily relinquished his keeping duties. He has also bowled and taken a wicket while a designated keeper.
Here we summarize the batting performance of these captains. The batting average is for those who batted for 20 innings:

The highest batting average for long-term captains is Andy Flower’s 49.28, followed by Mushfiqur and Dhoni in the low 40s. Paine and Sarfaraz are in the middle, while Moin Khan did not play enough to reach 20 innings.
Note the double centuries by Mushfiqur and Dhoni.
The dismissal rate, for those who fielded in at least 20 innings:

While Sarfaraz is no longer captain, his dismissal rate as captain is second only to that of Paine’s. But he and Paine have been dropped altogether from the team. Moin Khan and Mushfiqur are at the bottom, while Dhoni is in the middle.
Flower is the only one here who never made a stumping, while Dhoni has the most.
And finally we look at their performance as captains, considering the win/loss ratio. This is for those keepers who captained for at least 10 Tests:
(This table is still valid on 20 Oct 2022).

Interesting. We saw that Moin Khan performed worst as a keeper, but he did best as a captain. Similarly for FCM Alexander who did not do too well as a batsman. Next are Dhoni and Paine.
Taibu and Khaled Mashud failed to win a Test as captain, while Germon and Flower were just ahead with one Test win apiece.