I think that overstates the case against inflation.
Maybe so. But it also seems it has some difficult hurdles to overcome.
As noted, it's unclear whether spacetime is a thing. Is a center of gravity a thing?
Well, COG is a first temporal moment ... a mathematical approximation of a body as a rigid point. As one adds the second moment, third, and so on, the approximation draws closer to describing the body. Statistical moments can do the same thing, but in a different way.
Space is different, in that it defines a relationship between bodies rather than a description of a single body. Likewise with time.
But, I'll not argue with you, as I said it seems to be space-time is just an emergent property of things, not a thing in itself. IMO, for it to be a thing in itself, it must have a "here and not there" quality. For example, I can say I am here in my room, but not there in your room. So I am a thing.

I can't do that with space or time. I can't say space (or time) is here, but not there. When things are completely absent, time is absent. In an opposite sort of way, space is always and infinitely present. An absence of things doesn't make space absent.
The quantum nature of space-time really isn't understood at all. I'm amenable to the idea that it's all a quantum foam of interactions, but we have so little understanding of what that would mean it's hard to say anything.
It seems, then, you've not ruled out seeing it my way.
Stepping back from that brink, I think space-time, if you remove any singularities, would be some kind of smooth manifold that could be described as 'continuous'.
And that indicates to me even more that space-time is not a thing. Things reach a point where, if you divide them, it is no longer the same thing. But space or time divided is forever space or time no matter how much you divide them, even if you theoretically divide them beyond physical ability to measure them, they remain space and time.
If so, then when one says space-time is infinite, what is it that you are really saying is infinite? That question may be where we have to jump to the philosophy forum.