England captains traditionally resign when their batting drops off or when they've fallen out with everyone. Actual tactical acumen is considered optional, which is really odd when you consider how much Brearley is deified.
I think Root will probably go after this series, but you're quite right about the lack of alternatives. Burns would have been the other guy, long term county captain, but they've just dropped him.
I’m very bored and so have crunched some numbers. The following is the batting average of our last few long term captains in their final ten tests as captain, their career batting averages and the results of those last ten tests:
Joe Root (not including the current test)-90.7 (😲)/50 LLLWLDLDLL
Alastair Cook-51.6/45.4 WWLWWWLDLL
Andrew Strauss-67.2/40.9 LDLDWWWLLL
Michael Vaughan-38.8/41.4 LLDWWDWLD
Nasser Hussain-61.6/37.2 WDWWWLWWWL
Michael Atherton-44.8/37.7 LWLLLLWWWL
Interestingly all those listed above, with the exception of Vaughan, batted above their career averages in their final ten tests. The other glaring fact is that Root is batting at a phenomenal average and yet this has seemingly no effect on the performance of the team. He has to drop the captaincy and give somebody else a go.