All the NMAs I have done have been within a Bayesian framework, so I may be wrong here but I believe the frequentist implementations already use a multivariate framework in order to estimate the treatment contrasts so it isn't immediately clear to me how you would adapt that.
The only frequentist implementation I've see so far is: https://onlinelibrary.wiley.com/doi/full/10.1111/biom.12762
Some examples from Bayesian lit:
Repeated measures (uses fractional polynomials but multivariate likelihood): https://onlinelibrary.wiley.com/doi/pdf/10.1002/sim.6492
This paper: https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/1471-2288-14-92
This poster: https://pdfs.semanticscholar.org/b6d5/148ab793efb7b113a8ce001bb17f4eb74852.pdf
Other than that, pickings are slim although if you have any familiarity with WinBUGS you may be able to extend current code to multivariate likelihood. The fact that this hasn't really happened yet tells me that the big movers feel challenges aren't worth the pay off. If you look at the above poster in the context of multivariate pairwise meta-analysis lit, my guess would be:
You need covariance matrixes from each study and as a rule you will never have these.
Barring #1, you need to make assumptions which is fine but will make things harder for you. Alternatively, you can try to adapt the three level models out in the lit but you will notice they aren't cited often.
The improvement in precision that you can theoretically gain from MVNMA will only be realized if you have sufficient number of studies reporting all outcomes of interest; the outcomes are highly correlated; and you ALSO have a meaningful number of studies that don't report some outcomes. This is where all the borrowing strength talk comes into play in that you will essentially be imputing treatment effects. Even in the exemplar methods papers this only barely seems to work out.
If you're lucky enough to have reviewers understand #3, there's a good chance they will feel uncomfortable about whether they can trust the gain in precision you've earned.
So to answer your question directly: Yes it makes sense in principle, but Iit will be difficult to implement, require additional assumptions, will only make a difference in a minority of cases, and will be difficult to convince others to trust.
If you still want to go down this road please keep me posted on the results.
Edit: Missed a couple relevant refs first time around