Couldn't they just send out the largest sounding pulse in a given direction that they could safely generate, and if there is anything in that direction no matter if it's by Jupiter or pluto as long as the receiver is pointing that way it'd pick it up right? Or am I missing something about focusing the outgoing beam?
Focus is not the right term or concept. It is forming a beam, whose 3dB beam-width is usually called the beam-width. It'll keep covering a greater area as it expands out, but the angular spread is the same at all distances. Of course, if too far out you get too little power you can not see it or measure it too well.
So as you search space with radar you want some angular resolution that defines the direction uncertainty.
But you asked about radial distance. Yes, if it's really close to earth it could be dangerous, if further less so. Also when you detect asteroid you typically want to track their path, and you need some initial conditions and then updates, and position and velocity is what you might get: angular direction and range, and Doppler change for radial velocity.
However, you typically do better of course with optics, much greater wide angle angular resolution (just the diffraction limit). To locate in distance you triangulate.
Back to radar, keep in mind radar range (i.e., how far you can detect) depends on the power emitted (your 'largest sounding pulse'), the integration time (pulse width or Doppler integration time), your RF noise figure (how much internal RF noise), the size (really radar cross section) of the object, and most importantly proportional to 1/$R^4$ (two $R^2$'s, one each way). That means it dies pretty rapidly with distance. You can bounce radar off Mars or some of the other planets, but harder to detect off smaller bodies like asteroids.
So ,yes, as long as pointing right you'll get a return signal, but it may be way below your noise threshold.
Because narrower beam widths concentrate the power more, you see further, but cover less area searching a certain amount of time. Wider beams allow faster searches but you get a weaker return. So they write the equations and trade off on what they want to do. In terrestrial ground to air radar search radars usually have a wider beam width (or multiple narrower beam and multiple transmitter/receivers (and greater costs), while tracking radars are narrower beam widths, while for astrophysics for eg passive Radio and microwave receivers the sky is so big you typically use narrower beam widths and move the antennas around to search.
It's all a big trade off.