There are many Christians who believe this. It's called "cessationism," meaning that the charismatic gifts of the Spirit were given to the new Christians to help spread the Gospel, but once the Church was established, they "ceased."
I think the main problem with this is first what you said in that many believers clearly do have these gifts, and you can't say that everyone who claims to speak in tongues is making it up. Also, we still need to spread the Gospel! We still need to be able to show forth the power of God in the world and I don't think that just because the church is "established" that that means that the Spirit would no longer manifest Himself. The Church is not simply an institution, but it's living and breathing, and the gifts of the Spirit reflect that.
Plus, this is mainly an American idea, where Christians have become very lukewarm. In other parts of the world where Christianity is spreading and where there is persecution, you see these gifts manifested in amazing ways.
I can't speak for your church in particular, but I've been bothered by the same thing in different churches I've been a part of, and honestly I think it tends to be reflective of a loss of the radical Gospel vision and a tepid kind of faith in the congregation. But that's JMO.