Ritual Summoning is a summoning mechanic that existed since the beginning of Yugioh. It is unique in that the monsters do not reside in the Extra Deck.

Conceptual Design

The manga version of Ritual Summoning.

Rituals first appeared in Yugioh DM. In the manga, the player needs to evolve a specific monster on the field with tribute materials to perform a Ritual Summon. The Ritual Monsters were kept in a separate deck. NAS slightly altered the mechanic for the anime where a specific monster on the field is no longer a requirement. Despite this, when adapting rituals to real life, the mechanic became a main-deck mechanic.

History of Ritual Decks

Ritual decks barely saw any competitive success at first. In fact, Rituals only started getting regular support midway through the Arc-V era with Pre-Preparation of Rites. This boosted the mechanic's consistency, speed and recovery. It received further indirect buff due to Master Rules 4. However, Rituals still didn't see competitive success after all the buffs.

Misconceptions about Rituals

There are many misconceptions on why the mechanic fails. These are the most popular reasons:

• The inherent minus and consistency. People point to Nekroz and say Ritual Decks need to work like them to succeed. What they mean by this is excess Nekroz Ritual monsters don't get stuck in hand. This is false, as much of the new support addresses the consistency and card advantage issues without being identical to Nekroz.
  
• Dying to hand traps. Ritual Decks die to hand traps (e.g. Ash Blossom). However, that's the same as every other mechanic. Rituals have ways to play around hand traps.
  
• Slow. Ritual Summing doesn't have a one-card engine (e.g. Tour Guide From the Underworld, Wind Witch, etc.). However, Ritual mechanics do have a few one-card engines (e.g. Impcantation Chalislime and Gale Dogra). 

These issues are huge, but they're not defining why Rituals aren't succeeding competitively in the modern era. 

The Biggest Problem with Rituals

So what's the real problem with rituals? The sad truth is that many ritual monsters are mediocre.

Let's look at Nekroz. People often forget that Nekroz was powerful due to the monsters' potent effects and field presence. Unicore and Clausolas completely shut down the Extra Deck. Trishula, Brionac and Decisive Armor are removal. Gungnir, Trishula, and Valkyrus offer protection. They all have decent stats. Mix them with Djinn the Releaser of Rituals and you have a deck that shuts down every other meta deck.

This card in Nekroz shuts down every meta deck.

I want to clear up this misconception. Speed and consistency are essential, but they're nothing when the monster is lackluster. If Cyber Angel or Nephthys had the same consistency and recovery as Nekroz, they still wouldn't be high tier. This is due to these archetypes having ineffective monsters. You can give them a one-card combo and they still won't do anything (a common criticism for the Marincess archetype).

Ritual Summoning existed since the beginning, yet the scariest ritual monster is still Herald of Perfection which was released back in 2010. 

Why doesn't Konami just buff rituals?

Why isn't Konami printing out better Ritual Monsters? Here are the reasons why:

Supporting the Mechanic/Engine
In recent years, Konami seems to be designing Ritual support to make them a splashable engine rather than as an individual archetype. The Impcanation and Megalith archetypes were designed to help other Ritual decks. In 2018 and 2019, Demise/Ruin, Shinobirds, Nephthys, Litmus, and Vendread were made. These archetypes have little consistency compared to prior Ritual Archetypes. They were designed to be used alongside the new Ritual supports like Pre-Preparation of Rites and Impcanation.

A ritual released in 2019 with battle triggered effects.

Associating rituals with battle effects
Konami has almost always associated rituals with battle effects. We know Konami loves patterns when making support for certain mechanics. For example, Synchro supports usually facilitates extended combo plays and drawing. 

Hard to balance (most likely)
Just like Pendulums, Rituals either fail or become broken. Rituals are actually very consistent and fast thanks to their support. Giving them strong monsters will result in summoning them easily. So what do you do to counter that? You make a terrible monster (e.g. Megalith and Cyber Angel). The reverse is the same, you make an excellent Ritual Monster, and the consistency suffers (e.g. Amorphactor). 

Konami has tried to fix this issue by giving new rituals strange gimmicks (which failed). Black Luster and Vendread are two archetypes with a gimmick that involves inheriting effects from their ritual materials. Impcanation can be used to summon the monsters, but they'll be a vanilla beatstick.

Konami tried the gimmick again with the Ruin/Demise support. They locked away the Level 10 Rituals' full effects with weird conditions. Megalith is another ritual archetype that doesn't benefit from its Ritual Support due to the lack of a Ritual Spell. 

Anime Appearances

I have compiled a video containing all Ritual Summoning in the series. It has the least screentime compared to other mechanics.

https://vimeo.com/224357590

Why doesn't the anime feature more Ritual Summoning? There are several theories: 

1) Marketing.
The anime is made to sell products. No ritual products, no need to advertise it in the anime (e.g. 5D was made to sell Synchro).

2) Budget and Cost
The Ritual animation doesn't follow the standard stock animation in Arc-V, where it is uniquely based on the Ritual Spell. This means animating rituals cost more money and time. This also explains why Synchro appeared less than Xyz and Fusion in Arc-V as it needs more animation.

3) Ritual Design and Lore-wise
Rituals' designs tends to be more demonic and deity-like, which clashes with the anime's futuristic setting. 5D is a perfect indication of this. The Earthbound Immortal could have been Ritual Monsters, but the Turbo Duel premise prevented Rituals from appearing.

4) Writing a ritual duel is challenging.
The searching and adding will kill the suspense. The writers have tried to create suspense by hiding the cards added. There are times where the writers don't know how the mechanic works. For example, GX had Asuka reviving a Ritual Monster from the graveyard despite never Ritual Summoning it. Vrains had Soulburner Ritual Summoning illegally.

5) Poor attitude with Rituals
This is evident during the Arc-V era, where there was no Ritual Dimension. Only two Ritual users appeared and were quickly written off. Subtle jabs were directed at the mechanic where it is listed as inferior to Tribute Summoning in the LDS course. Minor characters were more amazed at FLIP Summon than the actual Ritual Summon when it first debuted.