There are several concepts in play here. First of all, you're correct that for stable orbits, lower orbits have both greater orbital speed and a shorter orbital period (higher angular velocity). When a satellite's orbit decays it can be approximated as slowly passing through lower stable orbits.
"The orbital velocity increases as the orbital radius decreases, meaning it needs a higher linear orbital velocity, not a lower one, to maintain its orbit."
It would indeed need a greater velocity to maintain a lower orbit, but by definition of "decay" it is not maintaining that orbit, only passing through it, so to speak.
When the satellite begins to enter the atmosphere air drag slows it more quickly and the speed of decay increases. This is a positive feedback system: the lower the orbit, the greater the drag AND the greater the drag the faster the orbit decays. When the instantaneous velocity vector of a falling body is parallel to any line intercepting that body and the (much) larger body that it's falling towards, then the falling body will not achieve even one more "orbit". This means that it's no longer useful to think of the body as being in orbit; it is now merely "falling."
In principle, a (very) round planet with no atmosphere could have a satellite orbit it only several feet off the ground!
In the case of the earth's gravity/atmosphere a satellite in decaying orbit loses its orbit status long before it gets near the ground (and generally before a person standing on the ground could see it well enough to estimate it's angle of approach.