A line-level crossover means biamping or triamping which definitely makes your system more complicated. For those of us with tube monoblocks, it means an additional 2 or even 4 amplifiers, each with its own power cord and mounting stand. If you obsess over coupling capacitors and other components, biamping also means doubling the number of parts to worry about.
My experience with biamping is limited to xover frequencies of 250Hz and below, but even with a relatively low xover frequency I found it was important for the amplifiers to be identical; otherwise there were obvious discontinuities in the sound. In particular, using a solid state amp on the bass and tubes for the mids and highs didn't work well at all. I am pretty sure the matching issue will be even more important with a higher xover like 1K.
There is also the issue of possible damage to delicate midrange and HF drivers. Ordinarily a small cone or compression driver will be protected from low frequency noise or amplifier turnon thumps by the xover capacitors. In a biamp setup the ampifier outputs are wired directly to the speaker and any LF signals like this go right to the driver without any filtering. With compression drivers costing so much, that's not a risk I want to take.
I have heard of people who have biamplified or triamplified systems who are very happy with the results. For my tastes, however, that's a move in the wrong direction. I prefer to simplify wherever possible.