It can be involving without having to add a cheezy story to it. Just have a career mode where you start off as a teenager, work your way through the amateurs and become a professional the way it's done in real life. Go thru the minor league tours in an attempt to qualify for the PGA tour properly not the way EA did it where you played a year and automatically qualified for the PGA.
You throw in additional factors like injuries from playing too many tourneys and money issues which may force you to play lesser tourneys because you can't afford entrance fees or the travel expenses to play at a certain venue. If you play a tourney in the U.S. and then want to cross the Atlantic to play the very next week, it may leave you susceptible to injury or just affect your attributes in conditioning because of the long travel, So if you became a PGA pro, you have to set your schedule where you takes weeks off at a time so you don't burn yourself out by playing weekly and across continents. Just like in real life.
You spend money for swing coaches, caddies and training facilities so the amount you win in a tourney becomes very important as it may not allow you to play the next tourney because you're nearly broke. You may be forced to go play a skins tournament somewhere for extra cash or be forced to go through qualifiers to play the bigger tourneys (like the US open) before you can get good enough to be a high ranking pro. And that's just off the top of my head. There lots of directions you can go with it that would have you playing for months.
If you paired that concept with some killer presentation like MLB the Show, this game would double its sales easily because it would have massive replayablilty.
Unfortunately EA focus thier resources on crap like the Night Club and Tiger Modes in the past games even though thier "telemetrics" tell them that 90% of the game is played offline. That means they must be playing career mode but get bored after a week because it is so shallow, No wonder sales are tanking.
I like what you are saying.... all of that sounds awesome! basically a dream game tho i doubt that will ever happen.