Yeah it's not a cheap approach but unless the water is coming in really fast it'll get the basement really dry. Key to have the drain and pump well below the footings otherwise yes the water will just be under the slab.
The issue with drain/pump solution is that at that point you're doing hydrological engineering.
There isn't a "perimeter" per se, because water is literally coming from depth (in most locals / soil types).
So drain/pump just creates drier 3d soil areas. Whether that's enough to attract all the surrounding water (at a high enough rate) that would otherwise be infiltrating depends on soil permeability (e.g. sandy or clay?).
A lot of the "handyman"-type "dry basement" shops have no idea about any of this, toss a generic solution in there, and then blame the gods if it doesn't work (and they've billed for time-consuming and expensive excavation).
Ah true. I live in a hilly region so the water issues tend to be rain water moving down hill (which is really mostly laterally). In that situation the exterior drain solution works well, totally dry basement. I don't know how common it is for water in a populated area to be coming from directly below but in that situation maybe it's a good place not to have a basement.
