No, really, there's not! It could work out either way depending on the economics.
With more frequent HW updates there's physically less time available for newer HW to garner enough of an installed base. Thus pubs will require games to target older HW for much longer.
In the best years of any generation of HW, consoles will generally sell anywhere between 10-15 million. In the first years however you're looking at sales of between 6-10 million consoles. That's with a traditional generation. Incremental iterative upgrades means both less time for building your installed base and less of an incentive for gamers to upgrade, so those annual sales figures will contract severely.
Putting new HW out more frequently whilst ensuring both forward and backwards compatibility will mean you sell less new hw during their launch years, not more.
You're judging how "old and crusty" hardware is based on whether there's new hardware out. That doesn't actually tell you anything about the limits of the hardware available, or say anything about what you as a game developer can acheive.
If instead you judge how "old and crusty" hardware is by its age (e.g. years) and by features and performance, then moving to more frequent, but less sizeable advances in hardware may
either force a lower baseline
or allow for a higher one depending upon a number of factors.
If the 360 had been "upgraded" every, say, three years then it is very, very likely that some games would have begun to target a higher baseline [than the launch 360 from 2005] before the 2014 or 2015 that most games actually did. The critical mass for support of a rasied baseline would likely have been reached earlier.
It's possible (likely?) that making PS360 the only console option for so many years meant that a crusty and old baseline was kept for far longer than needed - for at least some games.
If you look at PC land - where consoles go GPU and CPU shopping - frequent, incremental upgrades allowed some exclusive games (economics permitting) to move beyond the console baseline before PS4Bone was release.
As I say, it could work out either way, and be different at different points in the cycle, for diffcrent games with different economic realities. On PC every game is free to pick its baseline based on economic considerations.