I'd say "random access" is basically just a relative term. Tape, yes, is about the least random-access-friendly medium I can think of. Compared to that, HDDs are quite efficient at random access. DRAM then makes HDDs look like tape in comparison, but even that in turn has its own slight differences in behaviour between sequential and random access (row buffers, prefetching...).
but even that in turn has its own slight differences in behaviour between sequential and random access
It's not slight. Modern RAM has about 20x more bandwidth doing sequential reads vs. random reads. It's almost entirely due to the addressing setup times needed to access different parts of RAM (something I suspect most programmers don't even know exists).
Programmers should actually be treating RAM the same way they treat disks, if they want the best possible performance the hardware can deliver. You can treat the L3 cache the way you used to treat RAM—they're certainly big enough these days!
One of the problem I work on is completely random access memory bound, so to scale, we now use ridiculously cheap CPUs (AMD Kaveri chips if anyone cares) paired with DDR3 2400 DRAM, $100 motherboards, and an Infiniband fabric and switch to keep them talking to each other quickly. (We use Kaveri's because we also are doing GPU computation at the same time, and thanks to the slow RAM, the on-chip R7 GPU is more than fast enough to keep up.)
On the plus side, I get to spend way less time optimizing the code since the CPU just sits there idling most of the time, which is a nice change. :)
It's less a matter of age than environment -- there are still plenty of places using lots of tape. (It's not like we're talking about vacuum tubes or core memory here...)
Clarification: I didn't mean to imply that tape is dead, but that during the 15 years that I've been programming, tape drives have only been used in enterprise-level backup systems, and are not used as "archetypal example" in articles intended for a more general audience.
Tape is still in use today. Widely. I bet there are more companies archiving to tape than archiving to AWS. For most of the past 27 years it would be the dominant archival method.
I'm pretty sure that's tape, not HDD.