Can you use a router like a saw to cut wood?
It's entirely possible, and often reasonable. Without going as far as buying a CNC router (handy, but expensive) simple jigs and sleds permit cutting precisely circular holes (eat your heart out, jigsaws) and precisely straight edges (like a tablesaw with no need to use a jointer afterwards - indeed, many people with tablesaws use a router jig to joint boards if they don't also have a giant jointer.) A different sort of sled allows a router to act as a surface planer, as well, and a variant allows use as a curved surface planer.

There are, of course, compromises. You give up more waste in "saw" kerf. You may need to take multiple cuts at increasing depth or you risk overloading, bogging down, and snapping the bit.

From personal experience, you don't want a low-quality router with poor bearings. This is a tool where cheap can be expensive and expensive can be cheap. And it is certainly true that having more money in router bits than even a good router costs is perfectly normal.