I’d say it somewhat depends on whether you see yourself continuing to upgrade in the future, or if you’ve lived with the Denon for years and just want something better. If this is a one-time upgrade, then something like the Parasound integrated would be an excellent choice. It has a built-in phono stage and DAC, as well as two outputs to add subs if you ever decide to do that. You can find them used for under your budget and then spend a bit more on a turntable.
However, if (like almost all of us :)) you want to continue to seek better sound in the future, then separates make a lot of sense because you can upgrade the preamp or the amp as you continue on your hifi journey. You can certainly get great sound from separates at around $3,000. For example, the Aric Audio Unlimited II is $1350 new, and it’s an exceptional tubed preamp. You can pair that with something like a Parasound A23+ ($1699 new), and you’ll have a great system (sorry, I went $49 over your budget). And you can often find the A23+ used or discounted as a demo/open box at places like Audio Advisor. And there are plenty of very capable ss amps from reputable manufacturers that come up used here on Audiogon and at places like The Music Room that are well under your budget.
You can enjoy a setup like the Aric/Parasound for many years. And if (when) you get the upgrade bug, you have a lot of options to explore in either amps (tube amps, class d amps) or preamps.
Hope this helps, and enjoy the hunt for new gear.
Cheers, Scott