It is a mark of pride in the logic and justice of the Jewish legal system and an expression of Jewish identity and pride. It ensures that problems are dealt with internally. In many places and at many times it also reflected the unreliability of the non-Jewish courts.
The rule applies to individuals and also to Jewish institutions including synagogues. A synagogue that claims to follow Jewish tradition cannot pick and choose which traditions it will observe and honour, and if it refuses to accept a summons to a Jewish court its credibility is gravely undermined.
Yes, disputes involving synagogues have to do with their rabbis’ duties and terms of appointment and synagogue leaders have been heard to say that the rabbis on any Beth Din will automatically favour a rabbinical colleague and cannot be unbiased, though the fact is that when cases involving rabbis are heard by a Beth Din they will sometimes be decided against the rabbi; what matters are the facts of the case and the principles of halachah, not professional bias or personal favouritism.
Further, there are principles of Jewish legal procedure that give both sides considerable discretion when it comes to selecting the judges who will hear the case, and there cannot be grounds for complaint that lay people are being imposed upon or their rights trodden on or brushed aside.