I run a site with many posts (25,000 +) without a loss of backend speed (I do approx 25 posts per night) on a dedicated box without issue. In addition, my front end is VERY heavy with photos. I definitely buck the trend of lean-is-a-good home page.
Last January at around 15,000 posts my WP backend was running on a simple, cheap shared host plan. However, my front end got too busy (250k pageviews a month), so I had to switch to a dedicated machine.
On the shared hosting, any speed issues I had related to a) resource hogging plugins (had a mapping plugin I had to abandon), b) upticks in bots (I’ve had to ban IP addresses to slow russian and asian bots down, which made big differences on the share hosting pan), c) other hosts consuming resources on the box.
I’d be happy to provide more info if it would help.