It's true that you can't extend the number system to include something called "infinity" that would behave like a number. Just like you can't extend the number system to make division by zero make sense while keeping all the usual laws of arithmetic. But this doesn't lead to any philosophical implications like "infinity doesn't exist". In every context where it makes sense to talk about infinities without leading to contradictions, we want to do that, because it enriches our vocabulary and allows us to solve more problems.

Allow me to illustrate. Take the following problem stated in school geometry terms: A square is cut into triangles of equal area, now prove that their number is even. Interestingly, we don't yet know any elementary solution to this problem that can be stated in school math terms. The only solution known was discovered in the 1970s and relies on something called "p-adic numbers". What are those p-adic numbers you ask? They are weird number-like things that have infinitely many digits to the left of the decimal point. Freaky constructs without any counterpart in the real world, but you can still add them, multiply them and all that. Do such things "really exist"? Don't know, don't care. But they helped us solve a difficult problem and that's all that matters.