Mule deer are not hard to kill, hit them in the boiler room and they are going down.
I started with a 30 year old 30'06 that was given to me. When I decided to upgrade I moved on to a 7mm Rem Mag since I was also hunting elk. Now for deer I have moved on to a 25-06 unless I am also hunting elk and then I'll be packing my 340 Weatherby Mag.
Historically I've been a .30-06 guy, but the last few years I've really grown to love the 6.5 PRC. Last year I did some testing with that caliber and copper bullets....bomber combo!!!! This is my new go-to for most big game animals, including big bodied mule deer and elk. Love this combination.
Damn near every deer I I’ve killed in the last 40+ years was with my 25-06 most have been Blacktail but mule deer die just a quick.
Others I’ve used
.280AI
30-06
.300 wm
Another vote for 25-06! I built my first one in the mid 1960's when it was still a wildcat (.25 Neider). My Dad talked me out of it and I bought a new M77 Ruger in 25-06 in 1970. I have not used any other cartridge since then for deer, antelope or feral hog and I have taken way over a hundred of them.
I do have a couple of 30-06s, a .243 and a .300 Wby. My elk (and larger) choice is my .300. No need to choose anything else.
Every deer and elk I've ever killed in a modern firearm season, has been shot with my old 30.06 of Dad's that I sterted using at 12 years old (he officely gave it to me for my 16th birthday) or my 300 RUM that I won at a OHA banquet.