Is it really hard to make the phono section of preamps with built-in phono (at under $6K) as good as very good separate phonos at $1K ~ $3K?
Yes, it is really hard. Everything Mike says is true. In addition to the interconnect there is also the problem where for almost any given component the single greatest cost is the chassis. For integrated amps this pretty much overwhelms everything else to where you will almost always find better performance from a integrated amp than any separates you can find for the same total price- especially not with the extra interconnect and power cord factored in.
This is all still true with a phono stage, but with one very important difference. The phono stage is easily the highest gain most sensitive component in all of audio. Nothing else even comes close. So it can pay big benefits simply to get the phono stage circuit physically and electrically isolated from everything else going on in the amp.
This also allows for a much more effective and isolated power supply. Power supply might not seem such a big deal in a phono stage, only mV after all. But that is exactly why it is such a big deal. Even tiny fluctuations in power are a big deal when dealing with such micro voltages.
Finally there is the problem of budgeting. Every manufacturer has to look at the cost of every component part. Then they weigh that cost against the total package they want to deliver. The builder of a preamp or integrated amp knows every single customer will be using these. They cannot say the same about the phono stage. So when corners need to be cut, guess which one is likely to get cut first?
Now you start to understand why it is that whenever you read a reviewer comment about inboard phono stages being good they always go something like, "it is surprisingly good...for a built-in phono stage."
Specific to your question, take a Decware ZP3, and interconnect, under your $3k total, I think you will have a real hard time finding any integrated with a built-in phono stage for $6k the Decware will not totally wipe the floor with.