Regarding your question on perceived leadership. Its not your Job to know all the Details or know more than the functional experts in your cross-functional Team. Your Job is to steer and leverage thier Expertise to deliver.
It can come down to basic Project Management which at a first step is to get everyone together (Workshop/kickoff) to align around a common objective (strategy maybe). From there define together each fucntions contribution to the Overall objective/strategy.
I would first propose that you define the objective/strategy based on your Research/analsys of what's needed, the pressure test that with your Group as part of your kick-off. Alot of it Comes down to not only what your propose but how you gather Input. And in my experienes a great tool here is leverage your Workshop faciliation or presentations faciliations skills. Talk to your Boss on methods to collect Feedback.
Don't Forget off-line stakeholder Management, i.e. you 1:1 Meetings with your Team before the Overall Meeting to get insights/and pressure test any hypothesis you may have and get some insight into what they are thinking.
This is something I had to a few times over the past couple years as i've been leading a couple cross-functional Teams.
As for me, I've been in your boat, and have had to lead a brand Team where the I had to aling everyone on taking and pressure testing a certain direction for the brand - organizational structure did help in that it was defined that my function is an integrating fuction responsible for owning and Setting strategy for other functions to align against. However that means that all the other functions contribute to the strategy, they know the objective/strategy which we all defined together, and thus take responsiblity to operationilizing against the strategy which we all agreed on. I did the Standard Approach, aligned with bosses and pressure tested ideas, did off-line stakeholder Management to get views, did my homework, and organized my Workshops strategically to get the Information and alignment I Need. Pulsechecks and follow-ups are fundamental.
One tool you can use, that I did was take the most impacted stakeholder of your strategy (ie. your customer) for me it was "Sales". That Person become my "Sponsor" and he was my advocate when I started the process. He was also Long tenured and highly respected in the Company and I had him serve as my co-lead in many things. When you have endorsement of a well respected Team member or Boss, then you also get "soft but relevant" acknowledgement that you're the empowered leader.
So key Points, don't Forget your Project Management, practice good Workshop faciliating and presenting skills, dont Forget active-listening and asking clarifying questions, get a "Sponsor", and do your off-line stakeholder Management. Leverage Expertise of others and enjoy the ride. Leadership by influence is not easy and much harder than direct leadership, its a competency you will Need as you advance in your career.