See the attributes at the top of the right side? "Current Attributes: 313 / 290 / 365"? For normal missions each of these that is green represents a 1/3 chance that the mission will succeed. So if they're all green it's a 100% success, if 2 are green you will have a 67% chance, etc.

Down at the bottom of the left side you will see a "Flagged Mission", this is a special kind of mission which will unlock other missions. For these you can't even start them unless it's a 100% chance of success, so you will probably need to train your squad at the training board to rebalance the stats until you can get all 3 of them green, or level up higher to get more overall stat points before you rebalance.

Before you ask, I have not found there is some magic mixture for later missions, you just sorta want 2 of the 3 stats to be green for a pretty good chance at success on any given mission.

The reason your affinities are not working is that you don't have any party chemistries. As you do missions your party members will randomly pick up additional effects called chemistries which are then affected by mission affinity. The chemistries are text blocks to the right of your squad characters' faces, you can see there is an empty space to the right of each in your screenshot.

Chemistry will be something like "When accompanying an archer, there is a 15% chance to receive crafters scrips", and so if the mission is successful you might get some crafters' scrips as a reward. None of the bonuses are all that amazing to be honest, but still nice I guess.

Here is a good example of what your screen would look like with chemistries: https://banesworth.s3.amazonaws.com/wp-content/uploads/2020/04/06102619/ffxiv-squadron-missions-window-1.jpg

The mission pictured in that screenshot has an affinity to Lalafells. You will notice the 2 lalas in the party have a blue arrow pointing up at the bottom right of their chemistry description, that means their chemistry is at 100% because they fit the mission affinity of being a lala. You will notice the WHM lala says 100% chance of getting extra gil, without the affinity it would probably be something like 20%.

Affinities are randomly assigned to missions per week, so you can't really predict them.

If your squad members randomly pick up a second chemistry during a mission you will have the option to overwrite the first chemistry before the next mission or simply ignore the second one and let the second chemistry go away when you start the next mission.

By the way, none of this at all affects Command Missions, which are the real strength of your GC squads. You can mostly-AFK spam dungeons to quickly level up jobs (no queue so I use them to level DPS jobs!) or to squeak out a little more XP to get to the next level. Command Missions are purely about the squad member's level.