Most of the deer I've killed - mostly fallow, plus the odd red and sambar, I have shot with my little Browning Stainless Stalker. It is light, handy and accurate. I've had it for about 27 years.
I had a close encounter with a red stag one time. I was with another bloke, and as we came down into a little gully overgrown with lantana the stag was there, in cover. I smoothly sat, behind a log, but my mate hadn't seen the deer, until it bolted directly towards me. I thought for a brief instant that I'd be shooting upward as it jumped over me, but it balked at the last instant, not 5 yards away, and crashed back through the lantana until it emerged on the far side, full gallop across my front at about 70 yards. A bullet through the shoulderblade and spine rolled it like a rabbit, and that was that. I've killed a sambar galloping through blackberries at about 40 yards too, with the same rifle, and the odd fallow that was close in and determined not to be too.
I've also shot a bolting fallow at about 20 yards with a Marlin 1893. The 26" barrel was no hindrance - actually helped the swing-through. Even my Martini Cadet, 5.6x50 R, did for a doe lying down in cover from about 30 yards - I could only see her head and half the neck as she looked up at me, but it was enough for a quick shot before she could get up.
I've shot a few other deer close in, but a lot more pigs. Closest was no more than a yard off the muzzle, and closing fast. He meant to do me a serious mischief, had I not brained him.
The handiest rifle I have is probably my drilling. I just took it out for a walk last week in fact. 6lbs, 22" barrels, 1.5 -6x scope, and the immediate choice of 7x57R, buckshot or slug, it is ideal for really close cover, and proved as much last week. My double 9.3 is also pretty handy, and has plugged a good number of animals including those close in and moving. You save a lot of OAL with a break action.
I reckon the main criterion though is the rifle fits you, and you have confidence in it. It should come up like a shotgun, with the sights materialising before your eye. That allows you to get that bullet where it needs to be, fast. Provided that the bullet is suitable to the game that ability to get it there without hesitation tends to do the trick. .
