Finding a flat to rent with an L permit

I just received my L permit and I am disappointed as I was expecting a B type.

Does an L permit make finding a flat for rent more difficult in areas near to Zurich? I speak German on an almost native level, is this an advantage for the landlords?

Also are there any sites offering flats for rent other than homegate.ch?

For example a site which is only available in the official languages of Switzerland.

Thanks in advance for the help.