Are the Arleigh Burke-class destroyers weak in anti-ship capabilities?
They are somewhat, because that's not their main mission, while it is for the Sovremenyy. Udaloy is mostly an ASW ship (in Russian, it's not designated a destroyer but as a large ASW ship), and has neither particularly capable anti-ship nor anti-air systems, unless you count the lone Udaloy II. The Chinese ships expect to need their anti-ship missiles more than US ships because they don't expect to have strike aircraft and submarines readily available to handle it. US ships, by contrast, do, so anti-ship capability from surface combatants is mostly a reserve capability or for dealing with FACs and suicide boats which mostly wouldn't warrant a major anti-ship missile anyway. That said, the US is procuring the Norwegian Naval Strike Missile and the domestic LRASM. The latter will be deployed from both fighter aircraft and Mk 41 VLS on destroyers and cruisers. The new Constellation-class frigate will carry NSM, as will the the Independence-class LCS; there is talk of equipping the Zumwalt-class destroyer and Ticonderoga-class cruisers with it also. When LRASM goes live, the Burkes, Ticos and Zumwalts should all get it. For now, SM6 is a pretty scary anti-ship missile against anything that lacks Aegis-tier air defense - it's fast, has good guidance and hits with both a lot of kinetic energy and a fair amount of explosive with lots of fragments. True, it's not a proper semi-armor-piercing warhead, but that barely matters if you mostly just need to swat FACs and mission-kill larger ships long enough for a sub to come stuff a Mk 48 up their tailpipe.