An object on such a highly eccentric orbit would be classified as a long-period comet, which I mentioned. Yes, you can argue semantics about whether we might still call such an object an "asteroid" if it had a certain composition or primordial origin (which would be extremely unlikely for such an orbit). But there is no sharp division between asteroid and comment composition (it's a bimodal but smooth continuum) and this isn't relevant for the purposes of tracking such objects.
But that's not what I'm disputing. Asteroids routinely have their orbits mucked with by the planets and get thrown into (1) elliptical orbits, (2) non-planar orbits, and (3) ejected. In the first two cases, we aren't even looking for them because there is so much space that it's not worth trying.
Additionally, it is not true that we have found all dangerous NEO asteroids that are in ordinary orbits. NASA is still running a program whose goal is to find 90% of these [1]. For example, in just the past week they discovered a near-Earth object in excess of 1 km [2].
An asteroid in an elliptical orbit, for example, could be a civilization killer and might currently be undetectably far away right now.
We are missing a lot right now, and we need to spend money on much better surveys.