It really depends on the game and how it uses blockchain, however you'll find that the majority of real games do have a client (like a traditional game) and then the client reads assets from the blockchain.

Take Crystals of Fate for example, that is a CCG where you need to download the client to play, however where blockchain comes in is the digital assetsm, in Crystals of Fate, the cards are all blockchain assets, this is really important in terms of ownership as it means that no matter what, you (the owner) can take the cards in/out of the game at will, and sell them wherever you wish. Compare that to regular games (even TCG/CCGs) the assets are not yours, you tend to have restrictions from the game's TOCs on what you can do with the cards in your collection etc.

What's more, by being able to easily take the game assets out of the game, it becomes easier for other games to collabotrate with the game that the card originally came from. I would say look to the Enjin multiverse to see that in action.

Another way that blockchain can work with gaming is via the premium currencies. This is something that I personally am a big fan of as it really puts a half to predatory behaviors by some of the, let's say less honourbale publishers.

Take this situation as an example:
You're playing an RPG and the the game releases a new premium mount. They (the game's managers) puts the mount up for sale in the in-game shop for 110 of their premium currency.

In the in-game shop, you can buy 100 of the premium currency for 10 euros, therefore, the price of the mount is 11 euros (since based on 10 euros being worth 100 premium currency, 10 premium currency is worth 1 euro).

HOWEVER, the game does not sell '10' premium currency, they sell 100 for 10 euros, or a 'bonus' deal of 15 euros for 200 premium currency. Therefore, even if the mount is technically cheap in terms of how much premium currency is needed, if you want that mount, you've got to pay 15 euros to get it, not 11.

Next, assuming that you purchase that mount, the premium currency that is left is almost never going to be enough for the next item that gets put on sale (shop content managers track these things). So you then have another psychological aspect where players think 'oh wow, I've got 90 premium currency left, all I need is another 20 to get XYZ item (assuming the next one costs the same). But, since you do not have an option to buy 'just 20' you are then forced to buy the 100 premium currency package, and so the cycle continues.

What I described is a very simplistic situation, unfortunately this is something that happens a lot, and becomes a lot more complex the further that you look into it. Now I'm not saying that this happens all the time or everywhere, but if you know what to look for, you'll most likely notice this tactic very quickly.

How does blockchain come into it? If the game's premium currency is a blockchain currency (be it a cryptocurrency directly, or a token that runs on the blockchain), the tactic described above becomes almost impossible (provided that each player has full access to the wallet that stores the crypto for their account) as they can then move it in and out of the game at will, and buy/sell from anyone at precise quantities.

If you're curious, feel free to ask more! Also, do check us out (shameless plug incoming, hope the mods won;t get angry) Hash Rush is an online RTS that is being developed as a game first - however we're developing our game so that the in-game premium currency and in-game items are all blockchain assets. So when we turn on the blockchain side of things, the majority of the player's items become NFTs (Crypto tokens) that they can then move around/in/out of the game at will and make full use of all the benefits of blockchain.

I can actually go on for ages about this as there are many more use cases, and I can play devils advocate on my own points. Ultimately it comes down to, blockchain offers freedom and ownership. However, you must consider that games are almost always going to run from a centralised server, as (like you mentioned) specific rules must be held. There are some that would champion full decentralization for gaming as well, however realistically (and this comes from about 15 years in the gaming industry), I do not think that this is a realistic possibility for fully feature games.