I went ahead and took the plunge - got a Briggs 21R807-0069-G1 from Small Engine Warehouse. Physically, it was a direct bolt-in.
Murray uses a mounting plate that bolts to the engine and frame - the nuts on the inside of the frame rail channels are difficult to get to. I didn't have any trouble with them on removal (stuck on from age?), but I needed a Ford/Chevy distributor wrench to get the nuts up in place and hold them while tightening the plate bolts from above. My distributor wrench is an old KD or Cal-van that can be separated in the middle to attach a 3/8 ratchet - perfect for this job. The throttle cable fits perfectly. Since the exhaust is on the other side, I did have to drill/cut a hole on that side of the hood - I recommend the equivalent of at least a 3-1/2 hole. That will take out the lower hood/grille screw, so be prepared to add a couple of screws near the remaining screw on that side to replace the cut out screw. I also removed the muffler guard - that may not have been necessary.
Electrically, there are a couple of wrinkles, but not bad. The 21R807-0069-G1 has a 5-/3-amp alternator, just like the original engine that the original poster and I had. The alternator harness, carb solenoid, and magneto-kill wire all wire into a 6-pin connector, of which four pins are used. I believe the connector is a Molex 50-84-2060 - if you want the matching plug to wire up on the mower harness, it appears to be a Molex 50-84-1060. I found them on Waytek Wire. If you look at pics of this engine, notice that the alternator output harness does not go directly into the Molex - it plugs into a short pigtail that then goes to the Molex. Because the alternator is the same as my old, I just disconnected the 2-wire alternator harness plug from the short pigtail coming off the Molex 6-pin and plugged the alternator output harness into the 2-wire receiving plug in the mower harness. The black mag-kill wire goes into the Molex 6-pin that comes on the new engine. A pigtail is available that seems to use the same single-pin connector as the original, but I didn't want to take the engine cover, flywheel, etc apart. I elected to cut the mag-kill wire where it entered the 6- pin connector, cut the receiver connector off the yellow wire in the mower harness, and crimp on 1/4" spade terminals (heat shrink insulated) to make the mag-kill connection. That left only the feed for the carb-solenoid. On my mower's wiring diagram, I saw no place that appeared to be a switched, positive connection that would be on all the time the engine is on (thus eliminating the momentary starter relay trigger). I would have had to run a separate toggle/rocker switch to activate the solenoid. I elected instead to remove the solenoid, cut the tip off with my Dremel, and reinstall. I did need a thin-head 1/2" open-end wrench for that, as my regular 1/2" open was too thick (to my surprise).
I know this is long, but I wanted to share what I was able to do, since I had exactly the same scenario as the OP.