There are 2 metrics on which today's Claris is "better" than Dura-Ace 9000: range and low-speed performance (bigger cassette sprockets, and possibility to have gravel-ish sized chainrings). For the rest, the 10y Dura-Ace will remain lighter, more reactive, nicer to operate. Also characteristics like range might better for some, but not for everyone: for example a pro racer will prefer to have more speeds on a smaller range (and the range can be chosen depending the stage), to make sure they can select the most optimal ratio. Also, 10y ago, electronic transmission were available.
I can't talk specifically for Claris vs Dura-Ace, but I have an experience on the MTB side, having (had) a bike with 2019 Acera (one range above Altus, the Claris MTB equivalent) and a 15y old XT (one range below XTR, the MTB Dura-Ace equivalent). Even after 15y, the XT remain much nicer to operate: you need less effort to change gears, and they change instantly. Then not all features everything went to the lowest range: on one pressure the Acera could only change 2 gears (up), the XT 3. The index trigger works in both direction (pulling and pushing). Also the Acera as much more plastic parts, and after 1 or 2 years, it developed some play. So sometimes, I had to switch 2 gears and then go back to force the chain to move. It has been since replace by a Deore.
Now on which one to chose: it depends on your priorities. If you are not into cycling, the Claris will be easier and cheaper to maintain, and you probably won't use it enough to reach the point where play starts to develop. You will also appreciate the less race-oriented ratios (that also require training). If you are into cycling, you should know about the characteristics you are looking for, and the choice you'll consider would probably more 5y-old 105 vs 10y-old Dura-Ace.