Not that there isn't room for competition - there is. But the barriers are not what people expect. And the easy answers that coders tend to see aren't the real problems.
Most law firms are fully integrated into the MS Office ecosystem. Even with emailing things back and forth, the Track Changes feature in Word does a decent enough job. (Although transactional lawyers will say it's the bane of their existence.) More advanced firms will use a document management system to save and back up documents to the cloud, and such a system usually also has built-in version control. So, the main technical problem with lawyers emailing back and forth has to do with the fact that no standard exists for the industry for document management, so these document management systems don't talk to each other. Even if the firms are using the same system, the functionality that would allow collaboration between opposing lawyers is terrible, at best.
More importantly, the real barrier to innovations in the legal industry has to do with economics. Lawyers get paid by the hour, so there's almost no incentive to become more efficient. Large clients can demand efficiency in the form of slashing the final legal bill by up to 40-50% sometime.
Finally, even the smallest changes introduce professional liability risks, so lawyers tend to stick to well known and understood processes that have withstood the test of time. Unless there's a concrete, nontrivial incentive for a lawyer to try something new, you're not going to find one voluntarily sticking his neck out in the name of innovation.
If there's an industry that massive savings [to the client] can be had with technological innovations, the legal industry would undoubtedly be it. But for the reasons stated above, startups working in this space will run out of money long before they could introduce meaningful changes.
Read up on Atrium if you want to understand more about the Tech x Legal space: https://techcrunch.com/2020/03/03/atrium-shuts-down/
Plus for people in the first few years, there's pressure to meet a billable hour quota - work like this pads the numbers.
https://www.techcrunch.com/2020/03/03/atrium-shuts-down/amp/
[1]mkjung.com/contracts
A sample scenario is one of his senior colleagues takes a PDF and writes in some comments -- almost like a teacher grading an essay with a red pen, change this x to y, etc. This goes to my friend, who reviews, edits, and returns the draft by email. And there will be a lot of back and forth, plus other people making edits in real time.
watching this i can't help but notice how remarkably similar it is to programming, minus all the convenience of programming tools.
i don't know what the product or barriers are here, but these people could easily 10x if they knew about version control and markup languages.