I have the OM4T series cameras, and they are one of my favorites to use. I have seen a number of issues pop up in the cameras, the most common is a jammed wind lever with the mirror up. It can happen if you are shooting very quickly and winding on with a bit more force, it doesnt happen often but it can. You have already mentioned the battery issue with the OM4, you can test for new circuitry if you flip the lever to battery test, and wait 20 seconds or so, if the light and beeping sound turns off you are in the clear. Having to switch out batteries is kinda annoying in the field, not too bad if you have a 2nd camera, but you will loose that frame of film.
You should also check the spot, highlight, shadow, and memory functions, and their response to different lighting. On the side, you should also test the light up function, I have one camera that doesnt light up. Its pretty important if shooting in dim areas or at night.
Make sure the meter changes when you stop down the lens. I had one where the aperture ring under the bayonet lens mount was sticking and not giving a correct reading, it was fine when shooting as the camera uses the film when you press the button to meter, and it changes the exposure accordingly but it was annoying. I swapped out the whole bayonet mount, and shutter speed selector wheel, as it was that piece that was rubbing and pushing against the aperture ring(which is in turn connected to a string that goes into the camera) making it stick.
The shutter speed selector should be smooth when you turn it, sometimes it can get gunked up and become stiff, but its pretty easy to clean.
The F280 flash is a great accessory for the om4t, if you plan on using high speed sync flash it goes up to 1/2000th of a second. If you need it, it is a very good reason to get the OM4t model. If not the T32 flash is really nice and powerful for its size.
And again with any older camera, check the light seals around the back and as well where the mirror is.