The paper doesn't seem to have much discussion of theories that don't assume access to Universal Grammar, because the hypothesis favored by the authors does in fact assume that L2 learners have access to and use UG when learning an L2.
The introduction states:
The principal claim of the Full Transfer/Full Access hypothesis
(Schwartz and Sprouse, 1994) is that the initial state of L2 acquisition is the final state of Ll acquisition. We regard this as the conceptually simplest and most elegant view of transfer, as well as the null hypothesis, because in terms of cognitive architecture, it does
not require any additional stipulations to account for the phenome- non of L2 acquisition. Among approaches to transfer that assume
that L2 acquisition ’has access to’ (i.e., is constrained by) Universal
Grammar (UG), the Full Transfer/Full Access hypothesis stands in
opposition to the Minimal Trees hypothesis of Vainikka and YoungScholten (1994) and to the Weak Transfer hypothesis of Eubank (1993/94) (cf. also Eubank, 1994).
This initial state of the L2
system will have to change in light of TL input that cannot be generated by this grammar; that is, failure to assign a representation to
input data will force some sort of restructuring of the system (’grammar’), this restructuring drawing from options of UG (and hence the
term ’Full Access’).
The rest of the paper seems to be focused on the issue of transfer from the grammar of the L1 rather than on comparing Full Access theories to non-Full-Access theories (whatever the latter set may be).
I don't think your argument that "L2 learners would have access to UG just by dint of having a complete L1 grammar" is exactly right. Access to UG is needed for an L1 learner to acquire their L1 in childhood, but I guess that doesn't necessarily mean that the learner retains the same kind of functional access to UG in adulthood. I don't think it's logically necessary for the acquired complete L1 grammar to contain all of UG: it is commonly supposed that there are some things in UG that don't happen to show up in the grammars of certain particular languages.