Yes, bitcoin mining would be worse. Provably worse.
As computers get better, traditional applications, like databases, perform better and you can either serve more customers, use cheaper (and often more efficient hardware), or do nothing and use your outdated hardware.
With bitcoin, your enemy is not the performance of the raw work you are doing (you can hash better), but the fact that everyone else also has these increases. Because of that, the limiting factor is not performance, but cost, and so over time the average cost of the bitcoin network will not go down like the cost of centralized systems do.
Let's do a thought experiment: Let's imagine that instantly all computers get 100 times faster but it costs 10 times the energy to run these faster machines (not actually that unrealistic) and the cost becomes 10 times cheaper.
Traditional centralized banks can simply use a 10th of their existing hardware to have no energy-cost increase but a 10 times increase in customers they can serve. Sure, it's a bureaucracy, so it'll roll out slowly, but they're saving money too so maybe a little faster.
Bitcoin, everyone will simply buy 10 times as much hardware since it's cheaper and they want to get a bigger bit of the pie (mining reward) before everyone else manages to buy all their hardware. You now have an equilibrium in terms of equipment cost reached again, but suddenly the hash-rate is 1000 times more and the energy consumption is 100 times more.
The above might sound a little too theoretical, but if you look at the increase in hash rate and estimates for power usage of cpus, then gpus, and then asics, you'll see that in general power consumption for bitcoin constantly increases.
You're arguing that the existing cost of banks / the existing system is high because it's legacy. Your argument doesn't say "and it's constantly, and provably, going to waste more energy as hardware gets more cost-efficient". Even if you want to claim that banks are currently less efficient (which I doubt they are), I hope the above convinces you that bitcoin over time will constantly use more energy thus resulting in an obviously more wasteful result, while banks and other centralized systems can take advantage of improvements and thus become more cost and power efficient over time.