Hosting our Drupal site

TBH there are not many people visiting the forum with experience of large-scale Drupal hosting - you can find them elsewhere. I have done some relatively big sites but not with multiple servers. Nevertheless since no one else has commented, I will. My experience of Pantheon has been good for smaller sites. My feeling is that the cost of managed cloud services of the Pantheon and Acquia is probably worth it.

For doing your own server monitoring you are likely to use general tools such as Nagios, though possibly supplemented with Drupal-specific tools such as https://www.drupal.org/project/nagios. It is not clear to me from reading your question whether you want to replicate server or database. If you really want failover for database, presumably you will have to use master-master db replication because just syncing servers will not be dynamic enough, and master-slave replication will not give shared sessions table etc. (since slave will be for read-only), so you may be looking at clustering tools, as discussed here https://groups.drupal.org/node/451598.

There is a big part of the picture you do not mention: for larger sites there is a strong trend in the Drupal world, perhaps owing to the sheer complexity of the software, for continuous integration. The big Drupal hosting providers have a structure which can help with this. The whole question of development and deployment workflow has become central to keeping Drupal sites working, and the established Drupal hosts are helpful because they do provide some assistance with this, albeit you kind of have to work their way rather than your way.

Sorry not to give definitive answers. I am inclined to think that if you need and can afford a high-availability solution, it might be worth getting a bit of consultancy from heavy-hitting Drupal specialists such as Tag 1 Consulting, or 2bits, who can potentially help with performance as well.