I'm sort of confused - is the actual game the online mode or the offline mode?
Like... what should a normal progression of the game be? Do you beat the offline mode to beat the game? And is completion of the online mode even tracked as progress or is it more like 'hey I want to fight a monster online today' ?
Village - offline quests, solo only.
Port - quests with monster HP tuned for two players. Intended for multiplayer but still soloable, and available offline.
For me, I cleared village low rank, then got to HR2 in the port, then finished off the village quests, and then went back to the port. Beating village and getting your farm fully upgraded makes playing online easier since you no longer need to worry much about your traps, potions, bombs etc. I'm a bit of an anomaly and many players prefer to get some gear online when they run into trouble solo.
If you're 100% solo then I'd say clear village entirely before touching the port quests.
There is of course progress in the online quests. It even tracks which weapons you've cleared the arena battles with and there's a special reward for doing everything.
I'm not sure when you'd consider the game 'beaten'. You could finish the village quests but then you're missing out on an entire rank of monsters and gear by not doing the port stuff. You could say Dire Miralis is the final boss, but he's not even close to the hardest thing in the game. You could say HR999 with all quests clear is the end, but there's people out there who hit that and still play.