Can I activate and use InReach on 66i without syncing all tracks, waypoints, etc. to cloud?

AFAIK, you cannot prevent syncing entirely. However, you do have control over the visibility of your data to others. 

On sync, everything lands in the legacy "Library". Unless you choose otherwise, this is only visible to you (that is, to someone logged into the device's account in the app or on the explore.garmin.com web site). You can create additional data "collections" which reference subsets of the data in the library. In particular, at any given time, there is one "active" collection per device. Data recorded on the device lands in the active collection. 

You can also designate zero or more additional collections which are sync-d TO the device. That feature isn't really relevant to the privacy discussion.

The only place where anyone not logged into the account can see data is your MapShare page. You determine which collections (including the Library) are visible in MapShare. So you have control over what the rest of the world can see. In fact, you don't even have to enable MapShare if you don't want to. Do note that this provides useful information, though. For example, your family can use MapShare to see your location when you're in the back country.

There are also optional links to Facebook and Twitter. You don't have to use those if you don't want to.

I know the collections thing sounds excessively complicated. It does have its uses, especially for other devices (such as wearables) that have less capacity than a full-blown handheld like the 66i. You learn to manage your collections trip-by-trip (for example).

If you are concerned about data breaches (rather than controlling the public visibility of your data in the normal course of affairs), there really isn't much to be done. In that regard, it's worth noting that every inReach message you send or receive goes through (and is persistently stored on) Garmin servers. Messages also pass through the Iridium network and servers. And, depending on the delivery mechanism, an SMS gateway, cell phone provider and/or email provider. And GEOS servers if SOS is involved. That's the only way message delivery can work. Messages contain location information. So there's a certain amount of exposure no matter what.

Thank you for the prompt and detailed answer! It's very helpful.

Data breach is part of my concern but in general, why should I trust Garmin with my data? They may be well-intentioned, but accidents and bad actors happen. Working in software for years and years I am cautious about keeping data need to know even if it seems innocuous. Posting publicly is worse - consider the past issues with Strava. I don't use Facebook, Twitter, Instagram, etc. for similar reasons (not to mention that people and data are the product collected and sold). Understood that inReach messages will go through Iridium, Garmin, etc. network and servers. That's a trade-off I'm willing to selectively make for emergency communications. Alas, while i don't have a "Buried Treasure #1" type of waypoint, one never knows - and "Secret Fishing Hole #33" is possible. :-)

Thanks again. The 66i seems like a very nice device other than this concern and a bit of question on battery life, I like it.