My quick answer is yes - Blakies are generally freaked out by all the noise elk make. I have plenty of game cam videos of BTs running helter skelter just befor a herd of elk comes into view. Once the elk moved on, the deer returned to their normal habits. If you're seeing really fresh elk sign, then I would hunt areas adjacent to where the elk are grazing.
On the other hand, there's plenty of reports of trophy sized BT bucks hanging close to the an elk herd - and I'm talking about right in the mix of all those bigger animals. One guess for this behavior is that elk put out sentrys that watch major trails leading to the herd's location. If one of those sentry elk gets spooked by a predator, it runs straight to the herd - they know what that means - time to sneak o9ut or run wildly to get out of Dodge. Big bucks may be using them as predator alarms, or perhaps just attempting to use the large size of those animals to remain less visible/obvious to predators.
I used to concentrate my deer hunting close to roads believing that elk stay wll away from roads if at all possible. That tends to push deer to the peripheral areas of their territories, especially so for doe. Down here in SW WA, the doe populations tend to be less skittish around roads so they tend to be comfortable using those areas (since there are no elk there). My thinking was that if the former was true, then it would just be a matter of time, as the rut approached, untill bucks moved towards areas where the doe were. I believe that is true - I saw a lot of doe near roadside areas but I never did see a buck coming in to check on the girls. All in all, it's a real conundrum because you will find plenty of fresh deer sign mixed amongst the fresh elk sign which makes you believe that the deer are still there. All that being said, I'd recommend hunting away from fresh elk sign. It's a tough hunt for bucks if you're working close to the elk.