They can via primary radar, but the default displays secondary radar information (the transponder interrogation) as it tends to be significantly more precise. Surveillance radar has very coarse altitude resolution.
Thinking about it more, I've never tried asking ATC for an altitude check specifically from primary radar. It may well be that the system isn't set up for it, even if it's technically computable to some degree. When I'm talking coarse, I mean an altitude spread at range that may be measured in miles.
Surveillance radars have narrow beam widths in azimuth, but broad in altitude to allow them to quickly survey around them and get good enough awareness of where everyone is. There's always a trade-off between resolution and volume--some military tactical and approach radars have very precise azimuth and altitude capability, at the cost of scan volume/time. Mode C (and now, arguably, ADS-B) was the solution to this problem, allowing ATC radar to have both good position and altitude precision with an acceptable refresh rate, at a non-prohibitive cost.
You'd be surprised--unless your hypothetical binocular observer was really handy with stadiametric ranging, I imagine they'd be nowhere near a mile. They might be able to get a decent azimuth and elevation on a single aircraft, but that's a much different problem than determining position near-simultaneously for a hundred.