Is there a problem with this strategy that I’m missing?

Alternatives to password identification are required now.

Never re-using any passwords ever is fine advice, but then you are stuck using a password manager, which means you won’t be able to access your accounts from a public computer, or when using a borrowed phone.  And circumstances in which you find yourself without a computer/phone of your own (ie, your house burned down, you’ve been robbed, your hard disk has crashed, etc) may be exactly the circumstances in which you will need to access your accounts.  

Fundamentally, when it comes to password re-use, you have to balance security and convenience, the safety of nobody being able to hack your accounts with the possibility you’ll find yourself unable to access your accounts when you need them.  You also need to balance just how horrible it would be if someone hacked into each account — you may not particularly care if your Facebook is hacked, or you might regard that as almost as bad as having your bank account hacked. 

The other huge problem is how your email is the key to unlocking your entire online life — since almost every site allows password resets via email, if someone can hack into your email account, that gives them access to everything you’ve linked to that account, regardless of whether or not your password was unique.  Using one email for social media logins and a different email for your banking is a start (and then set the backup email address for resetting those accounts to yet a third email), but fundamentally there’s no good solution for this.

Generate a unique, random password for every site. The longer the better (as many characters as the field allows) . This is doubly true for your email accounts.
Do the same for “Security Questions”, which are notoriously weak. Don’t even try to answer them, just fill in the answers with another randomly generated password.

Use two-factor authentication whenever available, and pressure any and all sites you use to implement two-factor authentication if they haven’t already (Facebook and Gmail both implement two-factor authentication, so activate that and when possible use one of those to log in to other sites).

Suppose you choose one common English word out of a list of, say, 1000. Then you can be almost certain that your password is in the list of common passwords.

If, however, your password is, say, four English words out of that same list of 1000, then that list of common passwords would have to be 10^12 items long in order to contain your password with certainty. Or it would have to do an O(N^4) search, which is ludicrous. Use five words to be sure, though — 10^12 items is probably within reach real soon now. Using a 40-bit code for each entry, you’d only need 5 terabytes for that list.

So I do think the strength of your password is key (ha ha), and you can’t hope that every site everywhere is secure.

So I use OpenID for authentication whenever possible.

http://keepass.info

As to how easy it is, I’ve got my (admittedly slightly techie) retiree parents using it.

https://agilebits.com/onepassword

I think that a decently uncrackable source for ID and authentication could be something for governments to look at instead of fartling around trying to prevent kids from downloading porn and the like. 
(Except that like any government IT project it would be doomed to be a massive failure from inception till about 5-10 years later when everyone realises they’ve ironed out the bugs and it’s actually not bad.)