Note that
(%i12) fpprec:100;
(%i13) 1.0/3.0,bfloat;
(%o13) 3.33333333333333314829616256247390992939472198486328125b-1
So your example algsys(..), bfloat;
is merely doing the computation in ordinary floating point and
afterward converting it to bigfloats.
I think that algsys in your example is using allroots, which is
apparently unaffected by algepsilon and also unaffected
by fpprec. I don't know how algepsilon is supposed to work. The document
doesn't say. My guess is that
it doesn't work at all, since realroots and allroots, which are used, do
not look at it.