Don’t have sources on this but I also remember learning that 6000 doesn’t have to mean exactly 6000 and could be around there. Also I believe there is a gemora that talks about history being divided up to 2000 year periods and the last 2000 is the time of Mashiach
There have been many people who have predicted when mashiach will come and apparently often bad things have happened in those times.
Perhaps some sects (maybe just Lubavitch) have a strong emphasis on Mashiach. Yes it is an important concept but it is not our lifes purpose. Our job is to learn Torah and keep the Mitzvos. If 6000 comes and goes without mashiach then so be it. We want Mashiach to come so we could keep the Torah butter it is a means to an end.