Little Holger not only won the Masters at age 19 beating 4 top 20 players in a row, but he made the finals of 2 more Masters within 6 months and reached #4 in the rankings, as well as one point having beaten Alcaraz, Sinner, Zverev, Djokovic, etc , i.e. he actually had a great head2head record against top 20 players. Again, at one point early on he had beaten every top 10 player but Casper Rudd, the evil Norwegian. The buzz on him was legitimate and so yes, the expectations were quite high.
I know the Berdych comparisons refers in a general sense to early results /potential versus career results (IIRC Nadal detractors touted him the "Nadal killer") , but Berdych had very visible flaws from the get go, as primarily an offensive player his defense, return of serve and footwork was always suspect.
To me, Rune and FAA are more well rounded players with no exceptional glaring faults, (both their groundies and returns of serves are so solid, good instinctive net play, exceptional court coverage, touch, etc) who seem to have it all. but lack some inner mental fortitude to get to the very top. Certainly not a crime relative to the 250 players who plays tournaments on the circuit.
Dimitrov is a comparison I like, beautiful game on paper and on court, he's had a great consistent top 20 career , not quite top shelf elite, but always dangerous and so I think of him when looking at Rune and FAA.