You do not have the right to force your way into any group you see fit. You can apply to groups and they will decide out of the applicants they get, who they deem most suitable, most likely to help them complete the key on time or push it. Some groups will not care about the timer and just want to get a weekly done.
The leader absolutely has the right to decline you for lack of ilvl, lack of experience, not being the class they are looking for, not being the right armour type etc etc.
You of course are free to accept the first four people to apply to all your own keys and see how it goes. You can literally treat it like the group finder.
Forming groups is hit and miss but most want to succeed and to do that they will apply some common sense.