Sorry, no ideas off the top of my head, but i'll certainly give you moral support. I *HATE* players that do that!!!! One of my VCRs shows "Play" large enough to fill most of the screen for about 8 seconds. Then that disappears and a small version shows up in the upper left corner for another 10 seconds or so. What the heck???? Are they afraid that some user might not realized they pressed the play button? Even my best, very expensive, hi-fi VCR shows a little grey transport icon in the corner for about 5 seconds. Completely unnecessary and completely unacceptable, especially since the display on the front of the VCR also says "Play" or "Stop" or "FF", etc. too.
At least with the bad VCR i found a menu option that turns the display off, but it's 3 levels down in the menu and i have to do it every time i turn the VCR on. 'Twas very annoying back in my VHS dubbing days.
I several Panasonic players, 2 or 3 years old, that I use for theatrical projections. The Play and Chapter Skip buttons don't put anything on the screen, but the Pause button does.
If you find some that don't have the on-screen junk, but aren't multi-zone, search around the Internet and see if you can find a hack that defeats the zone garbage. Some can be hacked, and some can't. For those that can, some of the hacks are flakey and some are pretty reliable, so you'll have to take that into consideration as well.
Don't know about the zone stuff, but I have a panasonic dvd player about 3 years old that lets you disable the screen messages in a menu. When you shop around, look at the options menus