I'd say that yes, a game as the Ottomans without Westernization is possible, particularly if you go Free Trade and get lots of income that way.
How soon to aim for westernization as the Ottomans depends. The Ottoman military units in the early-mid game are equal to and at some points superior to western ones, so in some ways it's advantageous to not modernize militarily. But, if you westernize technologically and hold off militarily, you'll have the +50% stability cost modifier, which can be a big pain as the Ottomans, since the Ottomans in general have pretty high stability costs. So, there's an argument to be made for not making westernization a top priority. Nevertheless, unless you own the trading game, you'll eventually fall behind technologically without westernizing.
In the current multiplayer games as the Ottomans, I westernized (technologically) around 1550. At the time, my tech level averaged about 11, and European tech levels averaged just under 18. You could likely westernize a few decades sooner if focused on it, and with a focus more on the economy than expansion, you could stay closer in tech. But despite the overall gap, I beat all the European powers to Land Tech 18, so militarily it was possible to stay relevant despite a lack of a tech focus. IMO the Ottomans are also a bit better off than Russia in techs. Russia is in a marginally better tech group by tech cost, but the Ottomans have equal-or-better sliders (depending on Novgorod v. Muscowy), better starting techs and city improvements, and richer lands.
At the Ottomans, also be aware that the partially-westernized 50% stability penalty and a 50% overextension penalty
really adds up. The Ottomans already start with +20% stability costs from Jizya, and have several other generally-beneficial decisions that increase stability costs. I've spent many a year investing solely in stabilty, even with adopting Church Attendance Duty.
So, I disagree with Angst (who for so many years was known as lord_joakim! It is a strange adjustment). For the easiest time, always westernize (although not always at the soonest opportunity). But, for an added and perhaps historical challenge, you can often refrain from westernization and still be competitive. I westernized in the MP game mainly because I knew I'd regret it sooner or later with human opponents if I didn't. And I'd generally recommend it as the Ottomans. But it is optional. You can be competitive, albeit less so, without it.
Long-term goals?
- Shore up Asia Minor and the Balkans. Do what you can to make sure no big European power (*cough* Castille *cough* England *cough*) gets a foothold. Be aware that Defenders of the Fatih (*cough* Castille *cough* tend to be irksomely stubborn in peace negotiations.
- Make sure that you can always reinforce your European lands from your Asian lands, and vice versa. This can be by naval strength or control of the lands north of the Black Sea, but you don't want someone occupying all your European lands and not being able to reinforce them due to the Dardanelles
- Egypt and the Levant should likely be a target at some point. Decent land, stronger navies, and a bigger power base in general. North Africa is worth considering long-term for control of the Mediterranean
- You have a choice with the Hordes (usually - sometimes they give you no choice). You can get a lot of land for no infamy from them, but it takes a lot of work and, at least with the Timurids, will slow you down elsewhere early game. I went east to India in the MP game and secured the better part of India for the Ottomans - a rich land, and Persia is not bad, either. Going north could let you get nearly all of Siberia, as an alternate history. But both will sap resources from the west.
- Arabia is of debatable value, but is a nice area that would be of interest once Egypt's mostly secured.
- Of course, there's the option of going in to Europe - first Vienna, then perhaps Italy? I did this in my first EU2 game. Doing this too early might be anti-climactic, as Europe would be weak.
- Broadly speaking, success as the Ottomans is being stronger than the Austrians. Secondarily, also being stronger than any Russian power. Tertiarily, also being stronger than any Persian power.
The Ottomans aren't well-suited to New World colonization, but if they get to the Red Sea or Persian Gulf in decent time, can beat the Europeans to a good chunk of the East Indies. Being a first-rate colonizer would be an unconventional and somewhat challenging long-term goal.