I forgot to mention: If the old veneer is coated with kinda thickish laquer, you should try how sanding works; if it’s hard work, you could try a hot-air gun on it. You can then either scrape the old coating off while it’s hot, or burn it kinda crispy - it should make the sanding-down process a lot easier.
But wear proper masks for all those steps and make sure there’s enough air flow!
Edit: Be careful not to burn the wood if you try that method! :P
It’s gonna be hard and time-consuming work if you do it like I did, but the result is worth it.