Designs where the curved primary mirror is fixed, and steering is achieved via a moving flat mirror (siderostat) in front of it (or a set of flats) have been done before. One example is the Pfund telescope. The main design constraint that led to the invention of the Pfund telescope was the need to have a focal point which is fixed in space and easily accessible.
Here's a variation of that design; the need for the fixed focal plane is driven by accessibility concerns here (the user cannot follow a moving focal plane):
Also see the related concept of heliostat - same idea, just applied to sunlight. A moving flat reflects sunlight in a fixed direction. Some solar power plants use heliostats.
So, in principle, fixed curved primary + moving flat is doable, and has been implemented a number of times. However, there are substantial issues with this design.
First off, the size and the quality of the flat mirror are not less than the size and quality of the parabolic primary. Let's say you build a liquid mirror telescope because it's easy. Then you have to make a flat mirror of the same diameter, with the same precision, and all the ease of construction is now lost. Flat mirrors are not easier to make than parabolic. In fact, as a mirror maker, I prefer to work on curved catopters - in some ways they are more predictable during manufacturing than large, high precision flats.
There are further issues down the road for the liquid mirror telescope. You would actually need 2 flats, both very large, both very precise. Both would have to be suspended in the air, above the liquid primary, and supported by complex mirror cells just like curved primaries in any telescope. Overall, it will be more difficult to make such a telescope, not less.
So why is the Pfund design used? Not for ease of manufacturing; as I said above, it's actually more difficult to make it than a regular scope. But in some situations you just need that fixed focal plane for some reason. In that case, you don't have much of a choice. Of course, it's not a design that scales well.
I am assuming a large plain mirror is much cheaper than a parabolic
mirror of the same size.
That was the incorrect assumption. Once that's taken care of, everything else should be clear.