"why should we be ruled or even influenced by a bunch of mystics in the 21st century?"

We don't need an explanation about the mystic bit because we won't accept it anyway. We observe that they have a track record in engaging with a section of the population and then weigh up whether there is any benefit to the public as a whole in drawing on that. As with a political party or non-governmental advocacy group some will agree with their stance and some won't, and that will change as time goes on.

Just as the moment the church in the UK is taking a view on many issues that is on the left-liberal side of the political spectrum. All* it gets is a few votes in a chamber that has to defer to the one we elect.

* Not all of course, it can lobby in the background like many other organisations, but speed with which information can now be made public and transmitted makes it harder not to get noticed. One could argue that having 'representatives' of these groups in the House of Lords makes their stance and political involvement slightly more transparent - the old trick of putting someone on a 'commitee' where you can keep an eye on them.