The cost-effective impact and fundamental limits of infrastructure support with rate-limited wired backhaul links (i.e., GreenInfra support), directly connecting base stations (BSs), are analyzed in a large-scale hybrid network of unit node density, where multi-antenna BSs are deployed. We consider a general scenario such that the rate of each BS-to-BS link scales at an arbitrary rate relative to the number of randomly located wireless nodes, $n$. For the operating regimes with respect to the number of BSs and the number of antennas at each BS, we first analyze the minimum rate of each backhaul link, $C_{\textrm{BS}}$, required to guarantee the same throughput scaling as in the infinite-capacity backhaul link case. We then identify the operating regimes in which the required rate $C_{\textrm{BS}}$ scales slower than $n^ε$ for an arbitrarily small $ε>0$ (i.e., the regimes where $C_{\textrm{BS}}$ does not need to be infinitely large). We also show the case where our network with GreenInfra is fundamentally in the infrastructure-limited regime, in which the performance is limited by the rate of backhaul links. In addition, we derive a generalized throughput scaling law including the case where the rate of each backhaul link scales slower than $C_{\textrm{BS}}$. To validate the throughput scaling law for finite values of system parameters, numerical evaluation is also shown via computer simulations.