The easiest way to start investing in SIPs is to use an online app that supports eKYC. This will request the Registrar to send an OTP to your Aadhar linked phone number and when it is verified by the Aadhaar, you are all set. This can happen within minutes as you said!
Once eKYC is over the advisor will help you convert it to a full KYC. Even with the eKYC you can make substantial investments, upto Rs 50,000 per Mutual Fund House per year.