Also, it can absolutely be done with cheaper. My partner and I finished a 4800 mile trip last year. I used a second hand cyclocross bike that I got off craiglist for ~500 a few years ago. She had a road bike that she bought for 200 from the local bike project.
Getting the bags and racks etc won't be included. A good local bike shop would be my bet.
Instead spend the money in: good tyres (schwalwe, continental), paniers (ortlieb) and racks (tubus) and saddle (brooks) since this are the components that wear out really fast.
Gas stove does not work well in moderate cold climate. I discovered this the hard way. If the temp. Is gonna be about 0C buy a petrol stove. You can save a lot of money being pragmatic. I.e chopsticks are lighter than titanium fork.
If you travel alone buy a camping tent for two. You will need extra real state for your stuff.
Enjoy the adventure!
Go to an independent bike shop and explain what you want to do and your budget. They will be able to kit you out appropriately. For Canadian made bikes, my wife has a Devinci which, so far, has been great for rugged urban use with panniers and racks. I suspect that it would hold up great to touring travel.
It should be totally possible to buy and outfit a bike for $3k CAD
for the americans: a bike sold at canadian tire is often a $195 "bicycle shaped object" similar to what you'd see for sale at walmart.
some useful info on touring bike frames and components:
https://www.rei.com/learn/expert-advice/how-to-choose-a-bike...
https://www.cyclingabout.com/best-touring-bikes-you-can-buy-...
Classic choices are the Trek 520, the Surly Long Haul Trucker, and the Kona Sutra. REI has consistently impressed me with their offerings as well.
For the most part, frames get built with a carefully selected set of parts that balances reliability, repairability, and availability. Weight is really only a consideration for the wheels, and it's pretty far down the priority list even there.
Over 5000mi/8000km, everything is consumable, except hopefully the frame and associated components. I didn't blow up a hub over 7000 miles of touring, but I replaced chains, the cassette, the chainring that got the most wear, a couple of spokes, a rim, a saddle, and maybe a few other things. Oh, and several tires.
Mom and pop bike shop availability is clutch when you're miles from even a bigger town. A touring bike is not the bike to have the latest group set on, nonstandard spokes, or anything that you can't fix, replace, or make a workable substitution for in the boonies. AIUI, if you're touring internationally, you really want 559mm rims; 622mm is much less common outside of North America and Europe and finding tires can be a big problem.
You'll often find that frames get built up with an absolute hodgepodge of components that only makes sense if you consider all of the above, plus the fact that gearing runs more on the mountain bike end of the spectrum.
Components tend to run a generation or two behind bleeding edge, and manufacturers mix and match freely between road and mountain stuff as appropriate to get what they're looking for for durability and ergonomics. A good touring bike usually has mid-range components. Good enough to stay adjusted, beefy enough to hold up to a lot of miles with less maintenance than might be ideal.
All that to say, I haven't answered your question per-se, but I hope this helps you understand for yourself whether and answer put forward by a salesperson is a good one.
Most bike shops aren't making most of their money selling touring bikes. You run a very real chance of dealing with salespeople who are totally out of their depth when you shop for a touring bike. The ones that are both good and also honest will admit as much.
That may be touring-specific? I have tons of components on my regular bikes that last well over 5000 miles, including spokes/rims/wheels and saddles (derailleurs, shifters, pedals, cranks). Many tires even last 5000 miles, although not the lightweight racy ones I use.
Also, with heavy load (bike+luggage for me was usually in the 35-40 kg range) and more riding in rain than I would do when not on tour, leads to increased wear.
My worst failure was a cracked rim on a rear wheel that I should have replaced before the tour. It had started to wear down due to bad breaks. It was still rideable for another 100k to the nearest bikeshop (this was in southern Spain).
Going across the country you're gonna be hitting way more counties and states than you would even as someone who travels for cycling events. More importantly you're also not on a curated ride. Our club was top twenty and people agonized over the relative safety of certain back roads. Pavement quality, debris, traffic, blindspots, pets. Sometimes they got it right, sometimes wrong. It was an iterative process and some segments were only ever tried once (or twice if people forgot why we said no before).
On a cross country trip you don't have locals curating for you, so it's going to be more wear and tear.
That seems... unusual? I'm at 11,000 miles on my current road bike and I've replaced less than that even though I'm using components designed for weight, not durability. I'm on my fourth chain, third cassette, second rear tire (although both tires need replacement soon), and other than those the only things I've replaced are brake pads. Hauling a heavier load obviously increases wear and tear, but I wouldn't expect it to be that much.
I had 3 bikes, 2 of them did around 10k, 3rd one, well, I bought it at start of pandemic, then we moved WfH so it did much less.
Around 20% was riding around local forests, rest in city.
The first bike was literally like $300 (...probably closer to $400 now with inflation and all). I only replaced the saddle for more comfortable one but that's all. No disc brakes, literally the fanciest thing on it was that it had shocks on front.
But after finding job in big city and moving there I was using it for commute every day so after few years I decided "well, might as well buy something nice", bought one for I think like $2000, nice disc brakes, rear shock, lighter, "good" etc.
Comparing those two I went from mechanical rim brakes to hydraulic disk brakes and they did feel nicer, but in the end the rim brakes stopping power was just fine, it didn't feel as that much of an upgrade. Sure, a bit lighter to use but I could still yeet myself thru the front tire on rim brakes (probably). And rear wheel disc brake was entirely a waste, there is way less braking force needed on rear to lock up wheel.
In general I liked it but managed to break the frame (I was on max range of weight for that bike which probably didn't help), and decided to get something slighty cheaper, no rear shock (not much use for what I drive), cheaper accessories etc. (think it was about $1000). After a bit of riding turned out that only thing I missed was that tiny screw to adjust preload on brakes as the pads wear out. Drove few K so far, think some bearing in pedal broke (it's a bit wobbly), but not annoying enough yet for me to fix it.
In every case I had to replace sprockets as they use, replace the saddle for more comfortable one and install some accessory bits, replace the handlebars for one that fit me better and few smaller bit. But even the cheapo was reliable enough.
Generally the thing worth investing most is frame (as that will not change) and derailleur, as that will generally stay, brakes depending on riding (as I said, people driving hard off hills will probably have different needs than me driving some random forest dirt roads). Everything else you probably want to fit to your needs and you want to drive the bike for a bit to know exactly what you want.
Maybe "driving a bike" will get more popular as electric velocipedes continue to blur those lines.
I also find the word "bike" annoying, as it seems to mean both bicycle and motorbike, while in my native language there are entirely separate words with no common root (but funnily enough colloquial term for motorbike is same as "engine")
Also 50cc ones are called "engine bicycles" although that name isn't really used that often.
I would then immediately ensure everything works to your satisfaction, using this post-purchase assessment to not only bring the bike to the state it needs to be but to inform yourself on how your bike is put together, so that you can more easily repair her whilst on your trip.
I myself was able to pick up an incredible secondhand race bike built by a Japanese man who constructed such bikes for both the JP and US Olympic teams. Not only did I only spend a few hundred on it but I also was able to meet the man who built it and learn so much more about bikes and sprinting on them successfully than I had prior to the purchase.
TL;DR: Go secondhand, you’ll get a better bike and you’ll find people in the community who will help you become a better and more knowledgeable biker.