You use an amplifier to combat noise inserted by the cable
You increase the signal and the cable increases the noise.
Ideally, you're keeping the signal at a high enough level that noise doesn't take over and reduce the overall signal-to-noise ratio.
With a splitter, you're cutting signal strength AND noise equally, maintaining the SNR.
Since the HDHR is often placed right near the antenna, cable losses are minimal and amps are generally not needed.
Of course you need to remain above a certain minimum signal strength, but if you're that close to the threshold already, you have bigger problems with that channel
(i.e. the slightest weather disruptions will probably knock it out)
Sensitivity does not equate to signal strength. Sensitivity is the ability to distinguish signal from noise - no matter the volume.
If you take an AM/FM radio with borderline reception, does making it louder improve the sound? Does a quieter volume decrease quality of sound (to your ear)?
If you could somehow make your ear/brain more sensitive, you could distinguish more music/voice from that same radio than someone else whose ear/brain were not as sensitive.
It has little to do with volume. Thats what we mean by a more sensitive tuner.
If you take a 5th gen ATSC and 6th gen and put them side by side with an equal source, you'll find the 6th gen can pick up more channels (or pick up existing ones better)
They're receiving the same strength of signal, and same ratio of noise and signal, but one is able to better identify what is 'good' and what is 'bad'.