I went through this recently and completely de-Googled everything. I will try to answer your questions with a bit of my own experience.
What are my prospects of improving this situation?
Google has a cute little "take back my privacy" feature that is at best a socially awkward joke with an offensive punch line. It's just there to give advocates the warm fuzzy - don't think for a second it does any real good. The alternative therefore, is to move away entirely.
If you're going to move away from Google's prying eyes (and I fully support this) it comes with some sacrifices but is well worth it.
The best way to begin is to import your docs into something like Zoho or disroot hosting providers and are much more privacy friendly. Zoho has a very user-friendly application that will basically upload your entire Google Drive for you and make migration very, very easy. It also has the ability like Google Docs to collaborate online with others and edit via URL. Disroot is not as mature of an organization, but has very high regards for privacy and also is an Email provider. I bought a domain through GoDaddy as as registrar and moved my Email to protonmail, a Swiss company renowned for both their mail and VPN services (although I use VyprVPN instead.... for now). You can use a custom domain or keep the protonmail.com domain, if you'd like.
Ditch Chrome for Firefox and use DuckDuckGo as your search engine. This part might be the most crucial regarding Google's collection of information about you, even if you did not ask about this.
Finally, privacy from Google Docs is near worthless if you use an Android because they will accumulate most of your personal information anyway. You can install Copperhead OS on your Android phone which completely and utterly removes ALL google services and is built for the ultimate privacy (if not always convenience). You may opt for Waze or Nokia's "Here" instead of Google Maps. I traded Google Play for F-Droid and Aptoide. Finally, I traded Youtube for NewPipe and actually love the "play in background" feature.
Could I reasonably take out the data, encrypt it, and re-upload it to Google or another storage service to make it less likely that hackers, rogue Google employees, or any rogue authority can access it?
You could, but that would not serve much purpose. They still have your data and your encryption is probably no match for Google's engineers assuming they actually wanted it, and Google keeps a lot of metadata, meaning that while your individual documents 'might' be safe, your video consumption and music habits on youtube, driving history on maps, search history, doctors appointments, emails, etc is all kept as well assuming you use other google services.
If you REALLY want to erase ALL of your data, start looking in to and using tools found here and start doing a reverse google search on yourself. There are quite a few companies who aggregate and sell your data (persopo, radaris, etc), so you will need to go through them one by one and request that your records be deleted. It's time consuming, but if you're in IT security especially, it's well worth it.
Finally, consider how important social media is to you, because besides Google and the US government, Facebook is the biggest privacy violator out there, and LinkedIn and others are not far behind. (NOTE - I may be a hypocrite but I keep LinkedIn still. Tinfoil hatters gotta eat too!)
This may be extreme measures for you, and that's understandable - privacy and convenience has a trade off.
