tighter, punchier bass, clean power, room correction and more watts
Power could be sufficient in a small room if you set your speakers to small, which most do. Given the budget, a B&W 703 S3 (or equivalent brand) on sale or used + a generic class D module (VTV HypexNC252MP) will give you what you want for mixed movie/music.
The 703 will offer a significantly wider and taller sound stage while the amp will give you more power.
If you had said I want a larger, more delicate sound, then your original plan for an integrated amp with the same speakers works. The 703 does it in a natural way while the amp/preamp is more psycho-acoustic. I'm not sure you can find a separate pre/pro + amp for under $5k new.

