It's very possible to build a decent machine from wood, contrary to what you're hearing.
But it won't be easy.
For one, I've never really seen any plans for a "good" wood construction machine. Just about all plans for wood machines are designed to be easy to build, and inexpensive. Neither of which lend themselves to good machines. Building a good wooden machine is not easy, and requires a lot of knowledge, skill, and time.

You also mention that your on a budget. Trying to build an inexpensive machine is another way to end up with a poor machine. Good machine require good components. This is especially true of a wood machine, where high quality components can make up for inferior construction. Anywhere you try to save money, the quality of the machine decreases.

If you can weld, then your much better off with steel. It'll be faster, cheaper, and stronger than building with wood.