Q:

Can you explain gyroscopic precession using only Newton's three linear laws without applying their angular cousins?

Is there an intuitive approach to understand gyroscopic motion based on Newton's laws without passing through angular momentum conservation?

A:

"Intuitive" is a tricky word.  Most people find gyroscopic effects unintuitive no matter what we do.  And by far the most intuitive way to understand gyroscopic effects is through angular momentum conversion.  That reduces these effects to a handful of straight forward equations.
Fundamentally the motion of gyroscopes is based on momentum.  You wont be able to make sense of them without it.  Momentum can be viewed two major ways: linear and angular.  They're actually describing the same concept, but with different symmetries.  You can try to understand a  gyro using linear momentum, but because it isn't good at leveraging rotational symmetries, you will have a large number of integrals and sines and cosines involved.  Maybe that qualifies as intuitive for you, but my guess is it does not.  Gyros are not easy to understand in a linear sense.  We teach them in a rotational world with angular momentum because they are far easier to understand that way.

