So how does a non-pro team get experience? It's not cost-effective to field a team all year round without sponsors - but if you have been playing for years in pubs/friends I don't see why you can't give people a shot in May when exams are over or on the weekend so people with jobs can set aside time. As to the pro qualifier, oh no, the worst pro qualifier didn't get in - if you are the worst pro qualifier get better than the 20 other teams that are ahead of you.
There are plenty of open tournaments out there all year. I've participated in a few of them. You perform extremely well in all of those and you'll get noticed. Also you can go on IRC and ask pro teams for scrims. You do good against them on a regular basis and you'll get noticed also.
You could also join inhouse leagues, play very well there, and make connections.