First, in the specific case of the Il-96, four-engine passenger jets are pretty much dead because they have lower fuel efficiency and higher maintenance costs compared to twin-engine passenger jets.
As of October 30, 2022, there are all of three pending/backlogged orders for Boeing 747-8s and zero pending/backlogged orders for Airbus A380s
Even Illuyshin apparently is studying a two-engine model of the Il-96:
To reduce fuel consumption and maintenance costs, Ilyushin is studying a twin-engined development, powered by over 340 kN (76,000 lbf) Aviadvigatel PD-35s developed by 2025 from the PD-14, or foreign powerplants.
In comparison, there are 411 pending/backlogged orders for Boeing 777s, 484 pending/backlogged orders for Boeing 787s, and 420 pending backlogged/pending orders for Airbus A350s.
Second, technology. Russian technology - especially engine technology - is significantly behind US and European technology. Note the "powered by over 340 kN (76,000 lbf) Aviadvigatel PD-35s developed by 2025 from the PD-14, or foreign powerplants" in the above quote. Russian engines such as the Aviadvigatel PD-14 seem to have maximum thrusts in the 30,000 lbf/150 kN range. The GE-90 engines on 777s have been able to produce 80,000 lbf/350+ kN of thrust for almost 30 years, and can now reach 115,000 lbf/500+ kN of thrust.
Russian aviation is hoping they'll be able to almost triple the thrust of their engines. To get where GE was thirty years ago.
Third, economies of scale likely play a role. Russian passenger jet production has historically been minuscule compared to Airbus and Boeing. Only 175 or so Sukhoi Superjet 100s have been built, compared to literally thousands of Bombardier and Embraer regional jets.
To get some idea of the scale involved, there are over 4,000 pending/backlogged orders for Boeing 737 MAX jets and over 6,000 pending/backlogged orders for Airbus A320neo-family jets.
So if you need to order a few hundred passenger planes, you can pick from Airbus or Boeing for larger jets, or Bombardier or Embraer for smaller planes. And all of them have a demonstrated history of delivering, along with a large backlog of orders that help guarantee their continued production of whatever planes you order.
Russia doesn't have that demonstrated history, and doesn't have that backlog of orders helping to guarantee you'll get your planes. If you order 50 twin-engine Il-96s, you're hoping Russia will deliver them. But if you order 50 777s or A350s, you not only have much better guarantees of delivery based on history, there's also the knowledge that should Airbus or Boeing fail to deliver, your airline won't be the only one inconvenienced or worse.
So in a competitive market, ordering from Airbus or Boeing comes with both greater guarantees and effectively zero risk, at least when compared to the risk other airlines are operating under. No, it's not zero risk, but since just about every other airline is ordering from those builders, the comparative risk is lower. Going alone with a large Russian order would be very risky. If Boeing or Airbus fails, everyone's damaged. If you're the only one ordering from Russia and they fail, you are damaged.
And nevermind current political issues with trying to buy Russian-built passenger jets...