Three shot groups from a "cold" barrel are quite indicative of the performance of the rifle in the field.
Under hunting conditions, the first shot is the most critical and if the two follow up shots will go to the same point of aim 99.9% of the time the hunt will end well IF the shooter does their part.
I fully agree, there is a lot more information to be gained from ten shot strings, but like it or not, some rifles will never shoot a decent ten shot group.
The average shooter, with an off the shelf rifle should shoot to find how well the rifle maintains its "sweet spot" from a cold barrel to a warm or hot condition.
Some rifles are gems but most aren't.
If a shooter can afford to purchase several rifles and shoot all of them enough to establish which if any of them are gems, that's great. Most aren't.
The OP, developed a load and tested it on the bench.
That hardly guarantees the rifle will perform well in the field, under different conditions, rest points, wind etc.
You're right that he needs to do more work.
Many here are component starved because they aren't available or just can't afford them at present prices.
Only a few years ago, a person could purchase enough powder and primers for what it costs to purchase a 100 count box of primers, depending on which brand they choose to shoot.
OP, it sounds like you have a rifle that will shoot well for the first three shots. You don't mention if those shots were from a cold barrel or after you had fouled the bore and warmed up the barrel until it settled down.
The most important shots you will take on any hunt, are from a cold barrel, preferably sighted in at a temperature very close to the hunt conditions.
It's been my experience that the vast majority of rifles will maintain their accuracy from a barrel that's been previously fouled with a couple of shots.
Most rifles will "throw" the first shot from a cold barrel, AFTER CLEANING and maybe even a couple more.
What's important on a hunt is to foul the bore until it settles down, then take it hunting.
If you're one of those folks that insists on cleaning after every shooting session, even if it's only a few rounds to check POI to POA, then make sure you sight your rifle in for that condition.
If you happen to be lucky enough to have a rifle that will not change POI after the first couple of shots, you've won the lottery.
Most rifles will not shoot to the same point of impact after every cleaning.
A couple of us were discussing this a few years ago, and decided we would go to the range with a couple of rifles we felt confident with.
The point of this was to take proven handloads, shoot once, clean and shoot again, then repeat for ten shots on the same targets to determine if points of impact changed after each cleaning and if the rifles could be trusted to shoot consistently to the POA with the first shot after being cleaned.
Neither of my rifles would. The groups opened up to over four inches, from rifles that shoot sub moa, consistently, from a previously fouled bore.
Of the six rifles we shot, ONE, shot to POA consistently, after cleaning. It was a Savage, chambered for the 6.5cm. That rifle shot to POA whether it was fouled or clean. It was definitely a "jewel" Fugly, like all Savage bolt rifles but still a gem.
I have several hunting rifles that shoot very well, with reliable consistency from a bore that's been previously fouled, under all conditions. For me, that's about as good as I can hope for and it's also like winning the lottery.
I do have one "jewel" in the safe, it's a Tikka T3, off the shelf model, with the black plastic furniture and an excellent IOR Valdada scope, that will shoot within 1 moa of POA at 100 yds, from a freshly cleaned barrel. It will be passed on to someone that knows what's there and knows how to take care of it properly after my hunting days are over.
