What's the downside to injecting earlier?
You inject to a single-ended stage.
Most of the advancements in tremolo circuit design sought to eliminate the pumping effect when the trem is on but you're not playing. While you could argue this is an issue of background noise, there is some lower limit below which it is impossible to reduce the noise.
So the idea that designers came up with was to inject the tremolo in a balanced stage. Notice that Fender created the
Tremolux which injected the trem at the shared cathode resistor of the paraphase inverter. Doing this, the trem causes both push- and pull-sides to increase and decrease volume at the same time. But the output transformer only responds to a signal that's the difference of these two sides (hence push-pull); the common tremolo signal itself does not pass through the OT.
So if everything is perfect, there is no tremolo pumping with no guitar signal, but when you play, your guitar sound has its volume raised and lowered by the trem.
Same approach with output tube bias-vary, just a different injection point.
A single-ended stage can't do this. So the effort to inject late in the amp is all about reducing the pumping noise when you're not playing. The tradeoff is you typically need a much larger trem signal to overcome/impact the guitar signal at a later stage in the amp.