Do you ever feel like glitches can actually make the games MORE entertaining?

Depends on the type. There's glitches that enable you to do new things, those are great fun. Glitches that let you crack the face of the world open and let you peek inside, those can be cool too. Glitches that get in the way of normal gameplay suck though.

A lot of speed runs require glitches. Speed runs of games are so fun to watch because of the glitches, just because of the great lengths communities will go to discover glitches they can abuse to save a few seconds.

Absolutely! Take Cube World as an example. Cube World was pretty barebones when it first released (and frankly, still is). Visually the game was appealing, the combat was decent, there was a variety of (not especially varried) loot to collect... but after playing for a few hours you'd get the impression that you'd really done about everything that you could. And without any sort of end game, there was not much of a reason to continue with that gameplay loop.

Nevertheless, I continued playing that game well past warranted because of a particular bug, allowing rangers (one of the games classes) to move much faster when using their retreat skill followed by the glider (check out this video and this video to see what I mean). The glitch wasn't limitless (eventually you'd hit the ground). You couldn't use it to go anywhere (you inherited the momentum of the retreat skill, so you needed to keep going in that general direction). The glitch resulted in skilled, fast movement that I'd never experienced before in an open world game. Eventually I realized the gameplay loop was only an excuse for me to play -- my real reason was that glitch.

Of course, one of the few updates to Cube World patched that glitch out. That's when I lost interest in Cube World.

Tv Tropes has a page for glitches like that.

I'll just drop a few examples of some games and glitches that I've found have enhanced my enjoyment of the games. Loads of glitches I know basically just give you an easier game though, like in Dark Souls you can fairly easily dupe Homeward Bones (warps back to the bonfire), Souls (the currency/exp) and buff unbuffable weapons. These glitches are all used by speedrunners, but I don't think think they're necessarily good examples of glitches that really add something, instead basically removing limits to make the game easier.

One of the glitches can actually be fun to use though. It's used to swap one weapon's moveset to another's, so in speedrunning they use it to get a rapier's moveset (fast stabs) on a Ultra Greatsword with the high damage. But you can use it on any two weapons, and since Dark Souls is also a fun game to do challenge runs with, it basically gives you a lot of options to play with.

There's some simple stuff, like when I was younger I played a lot of Tony Hawk games. In THPS4, if you were to stand in a corner and do a trick into it (usually a kickflip is easiest), you could relatively easily clip through the wall. Basically just giving you a behind the scenes look at a lot of the levels.

Call of Duty: Modern Warfare 2 was the only game I played for a year after it'd come out, and by the time we had our hours in, there had been loads of glitches found. I don't remember all of it, but I remember pressing certain button combinations, swapping weapons etc. to get your shotgun to do a weird animation (like it spun around in your hand or something, don't remember it all). There was also a trick where if you used a throwing knife, held it, locked onto someone with it and then backed off, you could melee teleport knife him from literally 20 meters away (yes, it was worse than the Commando perk, but it was basically unusable online).
The thing that I remember most fondly though is in one of the levels (Quarry), if you were in a private match with a friend, you could (with friendly fire turned off) shoot each other with a shotgun, and by the knockback kick him up some stone steps that you wouldn't normally be able to get on to. Do that a few times, and you can reach the top of the level. Nothing major really, but I had fun dicking around with it back then.

I read somewhere that in the Starcraft remastering they tried to preserve all glitches so that competitive players who have exploited them would switch to the remastered version! I found that pretty cool

Tribes. There was a physics bug where if you mashed the spacebar on slopes, you'd get incredible amounts of speed. Combine that with rocket jumping/jetpack and you were moving faster than any object in that game. It became known as skiing, and became a core part of the gameplay; invalidating the use of most vehicles in match play. Super fun and made competitive play more challenging.

It was built in as a core mechanic in later games.

The first PC I played the original F.E.A.R. on had trouble with the physics engine. It would mean with sufficient force (like melee attacks or the nailgun-type weapon) to embed things (like enemy corpses) partly into walls.

I'd get distracted from the story for ages just seeing how much office furniture and dead clone soldiers I could kick into one wall.

One of the early Elder Scrolls games, Daggerfall, was absolutely full of this sort of thing. In the original 1990s version, you could make all sorts of game-breaking spells and items (permanent flight, etc). One particular glitch (or perhaps misguided design choice) was how the game decided on appropriate gear for NPCs. This was based on some value calculation involving the player's gear. It gave you to the option to start with a high-value Ebony Dagger, which meant the game would spawn NPCs with more valuable gear, allowing you to grind up high-quality gear very quickly.

A similar design choice was that you could make a custom class. It allowed armor and gear limitation by crafted material. There was no requirement that these be tiered in any way so that you could be forbidden to wear relatively crappy leather and chain, but totally fine wearing high-end Daedric plate.

A glitch is the reason Gandhi is a nuke-loving mad man in the Civilization series. At launch in Civ 6 siege towers never went obsolete so you could use them with your modern armour (end-game tanks) to attack cities.