Why Cannot Intel Compete Against ARM by Sticking with x86?
If Intel and AMD sticks with the x86 architecture, why would they have problems competing with ARM-based CPU manufacturers?

The problem with this question is that it is the wrong question. The question that should be asked instead is:
Why cannot ARM-based chips compete against Intel and AMD?
In the past, especially Intel has had a massive advantage controlling large foundries producing their chips, and their chips only. Because these semi-conductor foundries where at the leading edge making the smallest transistors, Intel could beat all the competition. Why did Intel have such a lead in manufacturing silicon chips? Because of volume. Intel made more chips than anybody else, and with volume follows economies of scale.

Yet they no longer have that advantage. The Taiwan Semiconductor Manufacturing Company (TSMC) has beaten Intel at their own game. And there was always just a question of time before that happens.
Read more: Is It Game Over for the x86 ISA and Intel?
It is the same reason why markets win over monopolies. Why planned economics at a national scale could not compete with free market economics. It is because multiple companies competing will in time beat a single company. Translated to the technology world, that means at platform consisting of multiple companies competing with each other will eventually win over a platform made up of only one company, or two companies AMD and Intel in this case.
There are no inherent weaknesses in ARM relative to x86 that makes it impossible to outcompete chips based on the x86 instruction-set architecture. ARM represent a whole platform of multiple companies competing. It is a platform the same way the PC is a platform or Android is a platform. We saw how the PC platform outcompeted all other single vendor platforms aside from Apple which got relegated to a niche.
Same has happened with Android. Android dominates the cell phone space because it is a platform with multiple companies.
Thus for ARM it has really just been a question of bidding time. This is the old story of the challengers starting at the bottom and moving up the value chain. We know how this story plays out because we have seen it happen again and again: Intel, ARM and the Innovators Dilemma.
At the moment far too many factors conspire against Intel, AMD, the x86 platform and the business model. One major factor is what I described in this story: Why Is Apple’s M1 Chip So Fast?

What I outline there is the fact that we are entering a System on a Chip (SoC) world, and that will not favor the dominant x86 business model, but it very much favors both ARM and RISC-V which are especially well suited for custom SoC solutions: Apple M1 foreshadows Rise of RISC-V.
ARM is Able to Cover Every Niche, Intel Isn’t
ARM competitor are able to specialize in multiple niches. I have written about this in depth before: Are Servers Next for Apple?
In that story I discuss the competitors to Apple such as Ampere and Amazon. They make chips tailor made for the sever market. This requires different optimization. Apple e.g. has specifically designed a chip well suited for the desktop market. It is a “fat” chip. Meaning it has lots of silicon on a single core with advance Out-of-Order Execution for maximum single core performance.
Ampere and Amazon in contrast opt for slim chips. Or rather chips where the cores are small so they can have lots of them on one silicon die. This gives a lot more hardware cores and superior performance in a server setting.
That is the kind of specialization you get when you deal with a platform consisting of multiple vendors all making their particular specialization. If you look at AMD e.g. they are using the same cores for chips made for gaming rigs or desktop PCs as the ones meant for servers. That is far from optimal.
Both Intel and AMD are to some degree forced to be generalists, but they are competing against a platform full of specialized companies which will make chips tailor made for particular use cases. Compare with with the difficulty of Apple competing with Android phones or PCs. They cannot make hundreds of different models. But in the PC and Android market you can get companies specializing on every possible desire and need.
How is Apple Able To Compete Against the PC Platform?
Now you could argue that Apple has held their own really well. In fact I constantly keep harping on about Apple’s strengths. So am I not being inconsistent here? Nope, because Apple benefit from tight vertical integration. The ability to fit software and hardware to each other. E.g. Apple add accelerators to their silicon and make sure to replace the implementation of frameworks developers already use to use these hardware accelerators. That way they can speed up Apple software.
Intel and AMD Lack Vertical Integration Advantages
Neither Intel or AMD enjoy that kind of vertical integration advantage. Intel used to have a similar kind of advantage by having their own semi conductor foundaries. But now they screwed that up and have fallen far behind TSMC. And this isn’t really a case where vertical integration gives the same advantage. Designing chips and controlling foundries doesn’t really give any synergies the way controlling software and hardware does.
Thus to go back to the original question:
Why Cannot Intel Compete Against ARM by Sticking with x86?
Because every company is rolling a dice every year. Random events will cause you to do better or worse relative to your competition. To win this dice roll every year you need to have a loaded dice. You need some inherent edge or advantage over the competition which they cannot replicate easily. Neither Intel nor AMD has that. They don’t control any key technologies which give them an edge. Quite the contrary.
The ARM instruction-set architecture (ISA) is better. It may not count a lot for larger chips, but it is still there. The M1 has demonstrated, that an ARM ISA gives advantage if you want to add lots of decoders to much more aggressive Out-of-Order Execution.
Thus sooner or later one of the guys at the ARM side of the fence will win the dice roll. Then Intel will start slipping, the key advantage in volume and hence superior manufacturing will be lost.