Is it really so simple as just getting to higher clock speeds than Intel did?
Yes, it is that simple. Up and including the 486 AMD's CPUs were developed close to Intels devices, supported by in detail information provided by Intel as well as reverse engineering. AMD adapted the design to their production process. This included low level changes in how certain circuits got implemented. Higher speeds are usually a result of that adaption.
Or, was there more to AMD's technical achievements even in these early years?
No in sense of functional structure. The CPUs were in all logic parts designed exactly like Intel's template. Same blocks, same path design, same registers, same micro code.
Yes as implementation of elements like transistors, resistors, capacitors did vary due different manufacturing technology. And that's an area substantial for gains in clock frequency and thus performance.
'Technical' (whatever that may be) advancement does not only come from processor design, but at least as much from chip design. And the later is a major component in clock frequencies a certain logical design can reach when made into a chip. And that has been an area were AMD was obvious ahead of Intel.
A Bit of AMD/Intel Licencing History
It's important to keep in mind that AMD and Intel agreed in 1976 to a cross licence and technology transfer lasting for 20 years (until 1995) which covers all their circuitry and micro code. It is that contract that made AMD a supplier of fully compatible CPUs from 8080 until and including the 80486 - Intel in turn got the 9511 FPU and other chip designs.
In 1982 they added another contract specifying technology transfer as primary means. So AMD had not only the rights to reverse engineer all Intel CPUs but as well the right to get all information needed direct without the need to reverse engineer. The 1982 contract got amended in 1984 were AMD agreed to not receive information about Intels new embedded measures for in-circuit emulation (ICE).
This was in turn the base for Intel's 1994 law suit against AMD and its AM486 product line, as AMD copied as well the micro code handling the ICE operation, which Intel claimed being a breach of the 1984 amendment. AMD's position was that the microcode was an integral hardware part of the 486 and thus covered by the base agreement.