This is a great example of Wall Street's continued but lessening snark towards blockchain and vicariously Bitcoin. That much of the complex system you know and revere and spent so long learning can be replaced by new fangled technology is still a hard pill to swallow. The snark is a coping mechanism.
1. https://www.bloomberg.com/view/articles/2017-01-31/mirror-tr...
2. https://upload.wikimedia.org/wikipedia/commons/thumb/9/94/Ga...
Furthermore, finance is not a homogenous group of luddites. Blockchains are being actively explored in the industry. There's even a consortium of large banks developing one - they're just doing it without all the decentralized sugar that bitcoin has.
Finally, as others have mentioned in this thread, a blockchain wouldn't have obviated this problem. The primary difficulty isn't identifying the owners of shares from three years ago, it's in correctly finding them and arranging for rightful reimbursement.
I think your characterization of the industry is misinformed and unfair.
NASDAQ - 11 million transactions per day.[1]
I can't see any issues here!
[1] - https://www.nasdaqtrader.com/Trader.aspx?id=DailyMarketSumma...
There are so many implementations. Many of these can process orders of magnitude more transactions than Bitcoin.
Bitcoin's maximum throughput is about 7 transactions per second. Superior blockchain implementations can generally achieve ~30 transactions per second for long term, effective operation.
Nasdaq's daily trades would require over 430 transactions per second. There are tradeoffs to be made (such as block size or integrity), but it's fundamentally not feasible to port our existing centralized ledger to a decentralized blockchain ledger.
This is just Nasdaq. Imagine trying to scale a decentralized blockchain to multiple markets, or to Visa's requirements, which are over 2000 transactions per second.
There is a variety of research on this topic:
http://fc16.ifca.ai/bitcoin/papers/CDE+16.pdf
http://www.tik.ee.ethz.ch/file/74bc987e6ab4a8478c04950616612...
You cannot scale up blockchains arbitrarily without making fundamental tradeoffs to increase their performance (for example, increasingly centralizing it).
Increasing the block size may indeed make a coin more centralised but we are talking about using cryptocurrency to replace outdated systems among banks, government agencies and brokers. In this case, it's not unreasonable to assume that the participants could afford to dedicate sufficient storage space to handle high transaction rates. There is no fixed limit on the transaction rate.