For starters you'll need people to manage the recording and storage, that means new people or retraining current employees.

It's already been successfully implemented in many courtrooms. In regards to costs, the new technology will probably save money over a greater period of time.

I don't think new technology should be avoided because of those mentioned issues. That argument hasn't prevented EZ pass toll booth technology from being implemented, for example. It's just the way progress is.

Even if it gets implemented, you have to deal with security and reliability. What happens if someone hacks in and gains access to restricted documents/recordings?

I'd personally argue that it's just a different nature of danger to security than paper records are. Both methods carry risks in that regard. We don't keep credit card transactions or stock market exchanges in paper form because of those risks. Regardless, the vast majority of court proceedings are not protected records. I'd say less than 1% involve records that need to be protected, if I had to estimate.

If the problem isn't detected immediately, even losing a few minutes of recording could be disastrous.

Like I argued above, is it much different than, say, the risk of a fire burning through the paper records?

Its pretty much the same reason why doctors and hospitals don't change what they are doing, even though it would make a much more significant impact on society. Giving doctors access to patient records would prevent loads of accidents from being made, for example. While some systems have sprung up, its still very early in its development.

From my understanding, those changes you say aren't happening are happening to some extent. I remember hearing a story on NPR about there being provisions in the recent Healthcare Affordability Act that required hospitals to transition towards more common formats and digital interchanbility among different hospitals. I don't have the link on me but I can track it down if you'd like.

Further, I believe the implementation of HIPAA laws in 1996 required common formats and codes for records across various hospitals in the country. It required changes and it was difficult, but I believe it has made healthcare better overall as a result.