Earlier quoted context omitted.
This is really a false dichotomy you don't have to use guid/uuid. I'm saying even if you use sortable auto increment numbers, stop storing them like numbers.
Note that reddit is currently generating base36 ids starting with z. You want everything to treat that as text? Sure. That text has been 6 characters long for ages, and it's about to hit 7. I personally expect to see more things break when that happens. The problem of overflow isn't special to numbers.
Suppose you store your ids as a 64 bit number in the database, that's plenty of room, we shouldn't need to put an alarm for when we're reaching the end (although operationally you might want to still) - it's 9,223,372,036,854,775,807 when signed and 18,446,744,073,709,551,615 unsigned. That's a fine, sensible decision to make at the database level. Now suppose you have an API that interacts with these ids in the database and exposes them down to users in JSON. Javascript (and JSON by extension) numbers aren't integers they're double precision floating point numbers -- quick pop quiz, without looking, what's the maximum representable integer before you lose integer precision? Do all of your APIs work with auto increment numbers (which should still be safe) or are some of them vendor supplied, like UPCs or ISBNs? Will your developers remember this when they make a new API or will they look at an old api and think "it's a number there, I should match it"? Do some vendors front pad their numbers with 1s (this happened to us)? There are a lot of variables here and one big juicy thing developers want and know other developers want - consistency of rules. So I'm offering an easy, consistent rule:
Store it however your database lets you store it compactly, know your limits, set alarms, stop apologizing for database lack of features, but MOST IMPORTANTLY: insist to your developers that its type in code is a variable length utf-8 string, regardless of it's contents. Overflows happen, you need to accommodate them at business logic levels all the time, but stop letting developers trust numbers as being self-bound. They're not numbers in the URL, they're utf-8 letters in a string. They're not numbers in JSON, they're utf-8 letters in a string. Give them an a function that verifies the format of a string and stores it however you want in the database, but stop treating IDs like numbers even if they're numeric.
That's my opinion. Not that ints are a bad db row type.