In general there are two levels of doing mitzvos, there is doing so because of קבלת עול – נעשה, and then doing so because of טעם ודעת – נשמע.
First and foremost we do Hashems rotzion, because that’s what Hashem command us to do, whether we understand the reason or not, whether it makes sense or not, we are בטול להקב”ה כעבד לאדון, whatever he says we do, with no exceptions.
Then there is נשמע understanding why Hashem command us do the mitzvos, what’s the reason for them, and what do we accomplish through doing them.
So yes 100% we so because Hashem told us do so (whatever we understand otherwise, whether we hasten Moshiachs arrival or not) but then when you understand why Hashem is commanding us do to so, we can get fully involved, with a chayus, and a geshmak, and אהבה.
We can’t just stay by the level of יראה we need to get to so also with אהבה.
The more we understand who Hashem is, why did he created yeddin, what’s his plan in the world, and what we accomplish through listening to him, the more we have a chayus in Torah and mitzvos. But that doesn’t take away the fact that we do the mitzvos Because Hashem told us so, it just adds, that not only you do it because you have קבלת עול but you also have a chayus on it.
So when people say we should mitzvos because it brings Moshiach closer, is not a wrong statement, (just like saying that we rest on shabbos because that’s what Hashem did). Because that’s what gievs him a chayus to do the mitzvos, that he has the zchus to bring Hashem more in this world thought Torah a mitzvos.
Every mitzvah we do, brings a gelui in this world, which eventually will bring the ultimate gelui, which will be when the beis hamikdash will be rebuilt.
When Moshiach comes, that’s when we made the complete dirah for Hashem.
And that happens through us doing mitzvos.
And this is more Hashems sake that he wants a dirah in this world, not for shechar and aoinesh.
Moshiach is not our personal gratification, even though we will gain a lot, it’s merely just because Hashem wanted so, and we have the zchus to fullfil him request.
As long as where on golus the shechina is in golus, שכינתא בגלותא. And we have the zchus to get the shechina out of golus.
If it’s not our business to understand the reason for mitzvos, why did Hashem tell us reasons for some of the mitzvos, why did the mekubolim and choikrim, be miayin in the reason for the creation of the world, gadlus Hashem, reason from mitzvos.
Again don’t get me wrong, we do mitzvahs out of קבלת עול, and that’s the first and foremost why we keep them, no matter what we accomplish, but to add and give a chayus to do them and not just do it with a dry spirit, there is a reason to know the reason behind them.
I would love to write and explain more, just impossible to type it all up. It’s pretty much Tanya and 600 page ספר מאמרים המשך תרס”ו from the fifth Lubavitcher Rebbe.