Correct up to October 15, 2017. The Test involving the ICC XI is not counted.
As mentioned earlier, there are only 4 Test players who played 5 or more Tests in which their team won all of them. As on Oct 15, 2017 they are:
None of them had particularly distinguished Test careers; for instance, Lord Hawke was a specialist batsman whose Test batting average was under 10. While GJ Bailey was a good limited-overs batsman he made only one fifty in Tests. They were fortunate in being in the right team at the right time-which was particularly true of Eldine Baptiste: http://www.espncricinfo.com/ci/content/player/51211.html
Not long ago Imran Khan (the present one, not the politician) also was on this list with 5 straight wins from his debut. But his 6th Test was drawn.
Adam Gilchrist had 15 straight wins from his debut in Nov 1999 to Feb 2001 (before the Australian juggernaut was halted by VVS, Dravid and Harbhajan). He finished with 72 wins, 11 losses and 12 draws in his 95 Tests (not counting the one against the ICC XI).
The runner-up is England’s Tim Bresnan who had 13 straight wins from his debut in May 2009 to May 2012 followed by a draw against the West Indies. He probably will not play in Tests again, and has 15 wins, 4 losses and 4 draws in his career of 23 Tests.
Here is a list of the 23 players whose Test careers had over 80 percent wins as on Oct 15, 2017:
The only current and near-current players are Jackson Bird (who is not in the current Australian team), Karun Nair, Wayne Parnell, Duanne Olivier and Dawid Malan.
Again, there are relatively few well-known players here (though it does include Loxton and Saggers from the 1948 Australians) besides Hodge and Lehmann from more recent times.