I cant see an issue myself in simply broadcasting his internet connection
That is, after all, the whole point, although rather badly put!! You are providing wireless access to a home network, and on that network is an internet gateway. It is not the internet connection that is being broadcast, just the SSID for the wifi network - and even that doesn't have to be broadcast.
As Keef says, encrypt the wifi. There have been loads of threads on here about all this - search the archives.
can I do a straight swap and use an old Belkin modem/router I have and set up a wireless network?
As long as the interfaces on the existing internet "modem" and the Belkin wifi router match up (e.g. RJ45 - RJ45) then there is no reason why not - assuming you configure it properly.
SD