I know that in 6x13 ‘Dreams Come True’, it is mentioned that Kurt and Blaine are in ‘Who’s Afraid of Virginia Woolf’ (which I assume in off-Broadway). This is also happening around the time that their baby is born, and I think that is going to be a tuning point in their careers.
I’ve always felt that fashion was a better fit for Kurt. Even though he has a great/unique voice for show tunes, I think his decision to pursue Broadway was definitely influenced by Rachel. And unfortunately, he would definitely struggle to get the “really good roles” on Broadway, and he was probably the target for most—if any—criticism for Virginia Woolf and that would have been draining for him.
Finally, the real reason I don’t want Kurt on Broadway is because it would constantly put him in competetion with Blaine and Rachel—and that's like constantly fighting an uphill battle. Kurt needs to focus his energy in an area which is unique to his abilities and where he can thrive and shine on his own.
I know that the show forgot about it, but I do think that Kurt continued his internship at Vogue until he got offered a permanent position. In which he dropped Broadway / Off-Broadway and pursued fashion (which he should have done from the start).
Whilst still working with Vogue, Kurt has to travel a lot internationally, which does test his marriage to Blaine and its hard being away from the children—but Kurt and Blaine have learned from experience, so they’ll be alright.
Kurt eventually goes solo and becomes a celebrity stylist. And he eventually gets his own PA.
Even though he is mostly a stylist, Kurt is willing to make some couture pieces for his most special clients. *cough* Mercedes *cough*
Kurt designs Mercedes’ wedding dress... and her three engagement dresses (massive photoshoot)... and her reception dress. This really skyrocketed Kurt’s career and its where he and Blaine get most of their money to afford their dream apartment.
Kurt also is one of Mercedes’ favourite stylists for the Grammys.
(I just want the Kurtcedes relationship back)
Wishful thinking, but at some point I think Kurt could be asked to be a feature designer for the Met Gala—definitely a highlight of his career.
As for Blaine; I think him pursuing Broadway was a bit of a weird choice for the show because he is more “Top 40″. Practically speaking, it was mostly to keep the NY storylines more contained.
However, I think a realistic explanation for Blaine pursuing Broadway/acting is because he loves performing, and he thrives in it.
Also, the only reason he wanted to go to NYADA was to get back with Kurt and to ensure their codependency (which was a problem in S5 but, fortunately, it was resolved somewhat in S6). Which is why NYU is a much better fit for Blaine; he is able to study a wider variety of areas and it would be more beneficial for his networking prowess.
Blaine does continue on Broadway a bit longer than Kurt. He is nominated for quite a few smaller awards and such, and they do attend a few Tony Award Ceremonies (mostly because the red carpet was on Kurt’s bucket list) but I don’t think Blaine ever got nominated for a Tony.
Because he is more family orientated, Blaine volunteers to be the stay-at-home dad when their baby is born. It gives him a chance to focus on his song writing career.
In hindsight, Blaine is very grateful for the roller coaster of a relationship he has had with Kurt because it has allowed him to experience every aspect of falling in and out of love = plenty of songwriting material.
Nearly all of Blaine’s songs are inspired by Kurt or their relationship; the good and the bad.
Blaine gains a lot of success and prestige as a songwriter; most notably collaborating with Elliott Starchild, and a one time collab with Mercedes later in their careers.
It all comes full circle when Blaine pulls a “Lin-Manuel Miranda” and writes an award-winning Broadway show.
Overall; Kurt is definitely more career orientated. Blaine is more family orientated.
For more headcanons: New Directions: Future Careers