A quick glance, in this case? Maybe. Someone coming in with a head concussion? On average (at a nearby university clinic) a hundred cranial CTs to find one abnormality. Many of those turning out harmless (similar to e.g. enlarged prostates, or calcified breast ducts). The biopsies to confirm, however, will cause a percentage of severe damage, which is why many screening tests are now postponed (e.g. recent media storm over breast screening recommendations being reduced, similar with more lax PSA watch and wait approaches). The number needed to treat (NNT) to save one life can sometimes be measured in dozens of unnecessary incontinence cases, later "unrelated" cancer (often eluding the statistics), thromboses after biopsies, etcetera.

Even when non-invasive tests are amenable to the situation at hand, you'd be surprised how many quick glances it takes to further rule out various unlikely hypotheses, or how long such quick glances can take in actuality. A short neural exam - and the mandatory documentation to go with it? Say 10 minutes, per patient. Rule out some additional unlikely hypotheses? Explain that to the gurneys filling the waiting room hallways. ER's are often crowded as is, additional waiting time will also kill patients. Efficient resource utilisation, using the resouces at hand, is crucial.

There's a famous med school saying that goes "when you hear hooves, think horses, not zebras". Much of the deviation from that rule is based on defensive medicine, which aims at avoiding costly lawsuits, a very poor surrogate marker for saving lives.

Pick your poison, but beware that there'll be "sob stories" either way, that could be avoided with other approaches.

Confirmatory "glances" only make sense to reach a certain certainty threshold in the main hypothesis. In the case of a girl coming in from a party on a friday night with slurred speech, I'd expect there to be thousands of (documented) "smell tests" - or breath alcohol measurements, to hold up in court, to catch that one case. And that's with a simple test available.