My suggestion is based on prior experience and hind sight.
Budget and lack of patience can really effect the shooters choice in firearms, but in the end it the proper choice will depend on the shooters skill set and the end goal of the rifle.
Some examples.
Shooter A is a young Marine E3 or maybe a civilian who started with Appleseed and really honed his skillset. Has a min wage job and all he can afford is a Remington 700, Savage, or Tikka. He saves up gets his rifle and buys a second hand Nightforce, IOR, Premier, MK4 or whatever scope. Smart choice, sure because of his skillset, he should be effective. The rifle may not print submoa with factory ammo, but he should be able to read wind, and put hits on targets at various ranges. The shooter has so much more potential to make long range shots but will be limited by his gear(based on many factory barrels and sloppy receivers).
Shooter B has money, and drops coin on a full blown out custom safe queen, every week he takes it to his indoor 100Y range, sets it up on a bench rest fixture and prints out 1/4 minute groups with factory ammo. Gotta take that rifle out on competition and holds up the line because he has no idea why he is not hitting at 300 with 10mph full value wind. Nice rifle but wasted on poor marksmanship
Shooter C where many guys fall into, they are eager to learn, but are not patient. Using budget to justify poor gear choices. Pick up a Rem 700 ADL, then upgrade the trigger, then upgrade again. Stock it in a HS precision, sell that for a B&C, then sell that and get a McMillan or Manners. Scope with a Millett, find out its shitty, get a Vortex PST, thats cool till the look though their buddies NF and sell the vortex and get that. Doesnt like 308 cuz 260 is where it is at sells the whole thing and starts all over again. A year later has over $3k in entry level rifles with entry level scopes because his "budget" did not allow him to drop $3-$4k on a precision setup in the first place.
Dont get me wrong, It is not about being a gun snob, but i fell into the shooter C category. I have a nice rifle and can be effective out to 700Y, just if I was to do it again, would have gone custom.
To answer your question, is the Tikka T3 a mediocre rifle, in the grand scheme of precision shooting, yes. Can you be effective with it, hell yes. The Tikka is a damn fine out of the box rifle, no need to upgrade anything (even with the factory composite stock). Put all your resources into the best scope you can afford, if it takes 6 months so be it. If all you can afford is the Tikka, the get it and scope it the best you can, then shoot it until the bore is smooth. Be shooter A not Shooter C.