I understand that you are using a milling cutter that has a Morse 2 shank. Some of those have drawbar threads, others do not. In older days, those cutters were widely used. They certainly can fall out if there is no drawbar and there is not a very good fit up between the adapter and cutter Morse taper shanks. Did you tap the cutter into the adapter with a soft head hammer? If not, try that and see if it helps. Another thing is to lightly stone the male taper lengthwise to remove any burs. A very tiny bur can keep the entire taper from fitting correctly. I have a couple new old stock (NOS) MT3 milling cutters without drawbar threads. I do not plan to use them on my mill even though I have an adapter like you have. I am afraid I might end up with the same problem, and ruin a job. I am saving them for use in the lathe tailstock if I need a quick flat bottomed hole, used after a smaller drill bit.
You can mark up the male taper with some High Spot Blue or equivalent, a very thin coat, transparent color. Install the male taper carefully, give it about a 90 degree twist, and carefully pull it out. Where the dye has been rubbed sideways you have contact, where the dye is unchanged you have no contact. Burs and dings are common problems, also spin marks in the female tapers. The burs and dings on the male taper can be approached with the High Spot blue and a fine stone used gently, just rub any high spots (you'll feel them) lengthwise, never crosswise, until they smooth out. Don't overdo it, you are trying to retain the original taper, or as much of it as you still can. The female taper can be checked for fit and any high spots can be dressed down with a MT reamer, gently but firmly, while trying to preserve the original taper, just removing the high spots. After doing some of that, you will take a lot better care of your tapers 8^). I have been there, I have done it...
Add the fact that cutting forces usually tend to pull a milling tool out of the spindle. So even with a taper in perfect condition and tightly inserted at the start, the tool will come out without a drawbar to hold it in.
My VN22l uses NMTB50 holders. I use a drill chuck with a Morse taper shank in an adapter. I know it's not the same type of forces on the taper but I've used some pretty big drills and the taper never pulled out. I seat the taper into the holder by dropping it in; then while holding the holder upside down dropping the assembly onto a wooden workbench from a height of about 6".
Am I understanding the OP correctly? The End Mill has an integral Morse Taper shank?
After you check the taper with some hi spot blue, and have it all clean, rub the taper down with some chalk and reseat it (if the taper looks good). That may buy you some bite. If the arbor has a tang, you might consider machining it off and drill/tap for a drawbolt, perhaps undersize like 3/8 -24 and a stepped drawbolt just for this setup.
Good to check for the B&S taper. It's not the same as morse. Detailed specs are in Machinery's Handbook.
Yes, you do understand correctly. Morse taper shank, no drawbar thread end mills. I have two NOS Putnam ones. They do have tangs. I'll post a picture after a while...
Edit: Added pic. I understand these were fairly common back in the day. The only place I would consider using them is in the tailstock of the lathe, following a pilot hole. One is 7/8", the other is 1", both MT3.
I believe that those tools are known as spotfacers to flatten the area around a hole drilled in a casting so the nut has a flat surface to bear against. force is applied against the taper not sideways bill