MSRP is a reference point. So is invoice. You shouldn't ignore either, but one is not a better reference point than the other. Maybe 20 years ago when you had to pay a publisher to get the invoice price, but now that invoice prices are freely available on the Internet, manufacturers have raised invoice prices faster than MSRP and closed the gap between invoice and MSRP. Dealer margin on new car sales now comes from backend rebates and volume incentives that aren't public. The only way to truly discover the bottom line retail price on a car is to present yourself as a serious buyer and get quotes from multiple dealers.
The good rule to live by is you should always be able to negotiate at least 10% off a new car. If one dealer isn't willing to negotiate, go to a different one. Helped loads of people negotiate prices for their new car and 10% is always achievable. The only brand where you can't is KIA - they just flat out don't negotiate, KIA UK sets prices and that's what you pay. The dealership might throw servicing or some accessories to sweeten the deal but there's nothing they can do about the price.
The only rule I know of is supply and demand. If there’s sufficient demand for a product, the seller has no reason to offer anything off. The proof will be that you won’t be able to find a seller willing to sell it.
Before the corona virus stuff, there were certain car models that you didn’t really negotiate much over, such as TRD 4Runners and Tacomas.
Tesla also doesn’t offer 10% off or haggle if I recall. I’m sure they would if they needed to move product, but I don’t know anyone who got any discounts for a Tesla.
The only good way to negotiate is to find a similar car at another dealer and work back and forth. Without another dealer willing to sell you a car for a better price, you have no leverage.
Adding to this, expand the search radius for a dealer by a couple hundred miles/kms. Check their online inventory to find the vehicle that you want, ideally by looking at the window sticker to compare inventory (website data is generally not reliable).
Many car dealers have APIs that will show the digital window sticker by passing a VIN number -- these are usually embedded on their websites and can be easily found.
Then, negotiate over the phone, reducing prices by 1-2k across each dealer with similar inventory to get a great price, below invoice.
I bought a 4Runner in 2007 for $22,500. The sticker on it said $32,000. Took 2 weeks of negotiation, and high gas prices at the time really helped ($5/gallon).
I would be surprised if someone was able to find one under $40k nowadays (maybe after COVID). You typically have to order them and wait a couple months to get it if you prefer any certain colors or trims.