Getting a PAN card and even updating stuff is quite seamless nowadays online.
https://www.tin-nsdl.com/services/pan/pan-index.html
Similarly with bank accounts. You call a bank like ICICI and they will send people to your house to open an account.
I don't know how a startup can make it better?
I used to see lot of "GET your pan card done" stickers around but haven't for a while now.