Will it work? Yes and no.
Yes, because in theory that approach is valid.
No, because for deep cuts. you should break it into multiple cuts, each cut a little deeper than the previous. So if you have a starter hole that is 1" deep, elongating the entire depth of the hole with a router in one pass will be difficult, because you will be removing an inch of material at a time, which will be very hard on the bit and the router. The bit will get very hot and you will probably scorch the wood, and it will wear out the bit (and the router) more quickly.
To elongate a starter hole, you should do multiple passes of elongations starting with a depth of 1/8" or so, and then going a little deeper each additional pass.
... turning perfectly good wood into firewood every day ... :smile3: