guess what, nobody uses the Black Scholes model anymore - quants use the Black Scholes pricing equation to recover derivative prices given an implied volatility parameter, and also to hedge (functional derivatives of pricing equation). and all of that 'deeper' thinking is usually ticked off in measure theory / functional analysis / stochastic analysis courses. you would have to be a downright fool to look at the properties but not the definition, the construction, etc, of the wiener process. but the properties follow from the definition. usually they ask because the properties can be exploited by a computer. do they ask you about properties of the wiener process? yes. do quants ask you to prove the existence of the wiener process? no. yet we use these objects every day and are often asked about them.

guess what though - all proofs of the existence of and the wiener process on use measure theory (and set theory) at some point. i am not even going to get into constructing the wiener process on. its obvious, right? if you are set on being constructive in mathematics, you will be very miserable. how do you know that exists? actually you dont. why? because quants use continuous time models, i.e. If we use your logic, you would have to present a constructive proof of the real numbers. sometimes even i think like that: 'this is not constructive'. one only needs to look at Hardy's comments about number theory as evidence (constructive proof!) of this stupidity. some people think as follows: if math topic X is not constructive, it is useless for application, therefore it is pointless to learn it. You are missing the purpose of measure theory.

Stochastic calculus with infinitesimals