The internet is simply a medium like any other media. New media does not replace old media. TV did not replace radio, DVDs did not put movie theaters out of business. Newspapers are dying because they failed to embrace online media, now that they're playing catch-up they will find a way to adapt & survive. For recruiting, the internet should simply be used as part of the overall media mix for business development & candidate sourcing.
And that's how exec search people can still do well - through research and using the internet to deliver a better product (candidate) & service (screening etc) that the internal sourcing people can do on their own - assuming they even have the time to do so.
Technology does not replace search firms. If you're a poor recruiter, technology will only help you do more of the wrong things, faster. If you're a good recruiter, it can give you competitive advantages and cost reduction by increasing efficiencies. Find a way to embrace new technology & make it part of what you have to offer in the way of services.