Re-installed Acrobat Pro X quits after about 10 seconds

Acrobat X is not supported on macOS 10.12. As was already explained, it came out long before Sierra, and Adobe would have needed a time machine in order to make it work with a future operating system. There is no way around that. Run an older version of Mac OS in a virtual machine, and then install Acrobat X on that. That's how I run older versions of Acrobat -  I use Parallels Desktop.

As to why it seemed to have worked before, it's very likely because it was already installed when you upgraded to Sierra. Based on my experience, chances are that it did not work 100%, but that you just did not notice any problems because you did not use all the features in Acrobat. Now that you tried a new install, you ran into a problem right away. That does not mean that it worked flawlessly before. There is no way around that.

Owners of old programs simply have problems with Mac El Capitan/Sierra... none of which is Adobe's fault... Adobe did not change your old program to make it stop working... Apple changed their operating system and THAT is why your old programs won't work, not because of anything Adobe did... Also... every time Apple issues an update, there is the risk that an old program that HAS been working will STOP working... bottom line, old programs require a matching old operating system