Start with your old Brio and see how it works out. If you find the sound muddying up on crescendos or it dulls down transients to make the music sound overall flatter, a power and current delivery upgrade is in order.
Before you automatically upgrade to the new Brio, listen to some other integrated amps in its price/performance niche, such as the Cambridge 640A v2, Music Hall 25.2, and especially, the Onkyo A-9555.
I say all this about the Onkyo from the perspective of using some respected high current power amps for the last dozen years. I have also helped friends and neighbors install affordable 2-channel systems using integrated amps from Cambridge and Creek. I used to power my 2-channel system with a Musical Fidelity integrated. Recently I heard the new Rega Brio powering Rega floorstanding speakers.
For transparency, speed, bandwidth, musicality, clarity, dynamic range at both ends of the loudness spectrum, and musical involvement, this Onkyo integrated smokes them all.
It should match well with your new RS6's and bring them alive. My Onkyo drives a pair of Mirage OMD-15s, which--like the RS6's--are rated at 91 dB efficiency, and have multiple mid/woofers in a ported enclosure. Although my VSP Labs TransMos 150 power amp weighs 50 lbs and is rated at 150/300 wpc into 8/4 ohms, the Onkyo integrated sounds bigger, clearer, more extended in the bass, and in general, more powerful.
At $450 from J&R or $474 from Amazon, it's pretty hard to beat. It DOES require a good 100 hr break-in before it shows what it's really about.