They might have had FTL drives to achieve this -- I highly doubt Pre-Imperial Humanity achieved it.
Humanity used warp drives and utilised warp travel prior to the Emperor's rise to power in the Unification Period.
but considering even the Eldar's capabilities in this are limited
I would hardly call the Webway 'limited' - especially in it's pre-fall state and in comparison to warp travel.
and the Tau have yet to achieve sizeable and quick means of transportation
That is more to do with their innate inability to grasp the warp and lack of individuals who are capable of navigating through it's depths.
the Astronomican cannot be his "divine will" made manifest.
It is the Emperor's will (divine or not) that allows the Astronomican to work. Without the Emperor it's just a bunch of psykers generating a psychic imprint, he is the one who guides it across the warp enabling it to be used as a navigational beacon.
assuming the portal to the warp is somehow kept closed
That's one massive assumption. And one that is quite frankly false. The Emperor is the one keeping the Terran Webway gate sealed.
than the Astronomican will continue to be lit, fueled by the will of ten-thousand psykers.
The Astronomican without the Emperor is worthless.
Now the High Lords will be utterly shocked that the Emperor is, in fact, not a god as they believed -- but that won't stop them from using faith in him to continue to rule.
How about the lore mentioned in Codex: Chaos Daemons
and Codex: Imperialis
that without the Emperor the galaxy gets consumed by the Realm of Chaos A.K.A. The Warp?