Randomizers have logic that restricts the way the items can be randomized. For example, in the default mode of the Link to the Past randomizer, you need the boots to get the item off the top of the bookshelf in the library, so that item will never be the boots. You need either the lamp or the fire rod to get through certain caves and sections of palaces, so you're guaranteed to be able to find either the fire rod or the lamp prior to having to complete those palaces.

Using glitches can help you "sequence break", to a certain extent - if you use a glitch called "fake flippers", you can reach certain areas of LttP that would normally require the flippers without having them. You'll never be required to do so to beat a seed in default mode, but doing so can save a ton of time if there's a required item blocked by the flippers and the flippers are a pain to get.

The LttP randomizer also includes "glitches required" modes, where varying levels of glitch execution might be required to complete the game, and the logic is relaxed accordingly.

A lot of effort goes into ensuring that you don't randomize the game into an unwinnable state. Many of these games are so well known by speedrunners and fans, that they know what items are required for each part of the game. The Link to the Past randomizer, IIRC, actually gives you the option to randomize the game in a way that requires no glitches, or randomize the game so that glitches may be needed.

In fact, it has multiple categories of glitches ("No Glitches," "Minor Glitches," "Overworld Glitches," and "Major Glitches", IIRC), defined by the ALttP speedrunning community, that the randomizer logic can take into account when placing the items throughout the game.

I thought this was a good introduction to that problem of unwinnable configurations.

https://www.youtube.com/watch?v=vGIDzGvsrV8

Sounds like a good opportunity to dust off those graph algorithms many of us learned in undergrad CS.

Randomizers are restricted to follow a tree branching logic to ensure that the game can be completed based on all accessible items at any given moment. For some games, there can be several difficulty tiers of logics. In harder logics, the game might only be beatable if the player is aware of an obscure glitch or nuance in the game engine.

The randomizers I have followed (Link to the Past, Final Fantasy) only choose viable randomizations.

They are aware of the items that are required to progress and make sure they are available appropriately.

The only up-to-date commonly used randomizers with meaningfully common impossible states I've seen are in Fire Emblem, which is difficult enough to begin with that unlucky changes to earlygame units and bosses can create a (near) unwinnable fight. You also have to deal with alternate loss conditions such as NPCs that cause game over on death and change to a class that is frailer than default or unable to move on the terrain type they spawn on.

The people making the randomizes usually have pretty good understanding of the game and it's obstacles. By building up a logical tree of what items are needed to reach which locations, you can make sure the randomized game is possible to clear.