Sarahkc

The dining packages offer a good value and I did the 5 night one last cruise mainly because I had sailed extensively with Royal and was looking for something different from the main dining room. Since this is your first cruise, I'd say go with the included options as you can always buy the package once onboard if the food isn't to your liking. 

Yup, every other restaurant has everything on the menu covered. Only exclusion would be the seafood tower and lobster at Chops.
Using menu from Brilliance again... Only general rule at Chops is only one entree can be ordered. Multiple appetizers, sides, and desserts are fine.
Chops: https://www.royalcaribbeanblog.com/menus/chops-grille-menu-brilliance-of-the-seas
Giovanni's: If it on the menu, order it. Don't think the have a single entree policy. 
Geovanni's menu on Adventure: https://www.royalcaribbeanblog.com/menus/giovannis-table-dinner-menu-adventure-of-the-seas
 
Just keep in mind the menus might not be exactly the same ship to ship. For the most part they will be very similar.

Since it's your first cruise, there is no harm in just going to the main dining room and skipping the dining packages. The food is still fine there. That gives you a chance to get the hang of things and see if you like it. If not, then give specialty dining a try next time. In all of our cruising history, we have only dined at a specialty restaurant a whole 3 times. And another tip, if you do decide to do specialty, your kids are only $10pp. DO NOT book them ahead of time. Just have them add the kids on once you get to the restaurant.

Only a la carte place I can think of is Izumi. With a $35 credit a person that can cover a lot.
Get a better idea of what everything costs check out: https://www.royalcaribbeanblog.com/menus/izumi-menu-brilliance-of-the-seas
Should also add Matt did a great review of Izumi on Brilliance as well: https://www.royalcaribbeanblog.com/2018/02/06/review-izumi-brilliance-of-the-seas

I'm in NO way disagreeing with you as you are entited to your opinion, as i stated in a post yesterday it's RC that makes the rules not the guests and as you yourself mentioned it's a suggested dress code not compulsory , HOWEVER you must realise that times change, you wont get a cruise like you used to nowadays! the days of the Love Boat style cruise are gone,  not because of the passengers but because of RC. Cruising used to be about doing very little except sit outside in the sun and read, then get dressed up to the nines for evening meals.
RC and all other cruiselines have made cruising more open for less affluent families meaning it's not just the snobs who get to cruise now. This means familes with children , young couples on honeymoon or family groups who are celebrating a special occasion can experience visiting new places that previously were maybe outwith their budget,  Cruiseline companies have realised that their Love Boat style guest will in time vanish, so they have been opened up to a new breed of cruiser because it's a business and they need to make money. By doing this they have had to change how ships are to keep these new cruisers coming back,  that  means ships have had to change to accomadate these HORDES ( your words) , Waterslides, Broadway shows, ice rinks to name a few have been included to cater for these new cruisers ( who may be teachers, nurses or even doctors).  would you want all of these attractions introduced removed as well, to return to the good old days of cruising? 
We should never judge a book by it's cover, like wise we should never judge someone by how they dress