Honestly, don't bother with a dGPU and get a 12th or 13th gen Intel Core chip with QSV. Intel quietly tuned it up to the point where it's faster than nVidia's NVENC engine even in the latest gen plus you don't have mess around with the uncap streams hack and you're transcoding through system RAM not dGPU RAM, so far less likely that your stream limit will be artificially constrained by memory limitations.
To answer the question you asked though, the nVidia NVENC is the best solution on a dGPU. It's performance is largely the same across the same board generation, with one exception in the GTX 10X0 series. The absolute cheapest card you can lay your hands on that has an NVENC engine is the 1050TI.
The caveat is the 1070 and 1080 have two NVENC engines. It will double max number of streams in theory, however in reality you're memory bound on those cards and it's more like a 33% bump.