suppose the court were compelled to take all cases that were appealed to it. for all the cases that it "did not want" to hear, it could simply mechanically vote to uphold the lower court decision. i'm ignoring the practical amount of work and time, of course, for this thought experiment. but after wasting the courts time, 5+ judges can say "we didn't want to hear this case, you wasted our time, the lower court was right, go away".
that could be done with a minimum of 5 judges of 9 in agreement to "let it ride".
in the current scheme, it does only take 4 judges to vote to hear a case. so presumably, if only 4 judges wish to hear the case and 5 are against hearing it, the slight minority must feel that having heard the case fully, one or more of the other 5 might change their mind and vote to overturn the existing verdict.
so excepting the situation where a 0-3 judge minority wishes to fully hear the case, i don't see much difference except a lot more work if they were faced to take cases that would surely be upheld anyway. it seems a pretty good conservative heuristic that if you can't even get 4 judges to hear the case, there's no way 5 are going to vote to overturn it.
so i think the article makes a lot more out of this than is warranted. i don't see it make much difference at all in comparison to things like the overall composition of the court or it's specific 9 personalities and that of the chief justice in particular.
EDIT: a different way of stating it: instead of saying a minimum of 4 judges must vote to take a case, flip it around and say: "if a super-majority of justices, 6 or more, vote to uphold the verdict after a preliminary review, it will not be heard in full"