You can transit without entering India. For this a Swiss gate agent meeting the arriving aircraft has to escort you to the international transfer desk and hand you over to the handling agent of Druk. The transfer desk usually has reps of all handling agents, if the Druk agent is not present they will call them to get you your boarding pass. Once you have the new boarding pass (duly marked 'DT' for direct transfer) you can go thru transit security and end up in the departure concourse. Swiss will let you board their flight to DEL because you have a visa.
The United/Vietjet transfer is tricky because United may deny boarding due to a missing visa and missing conjoint tickets. You can talk your way into it, but should assume things are not going to work out. The overworked/underpaid United agent in EWR will revert to his/her computer screen and see what it says.
The BoI guideline on the matter:
https://boi.gov.in/boi/public/conten...e-7c405e05303f
Persons in direct transit through India:
1.Travel by Air:
A Transit Visa is not required by a foreigner passing through, in direct transit by air, and travelling onward through India, provided he/she does not leave the specified precincts of the airport.
The PK sensitivity comes from the things David Headley (American with PK heritage) did in 2006:
https://en.wikipedia.org/wiki/David_Headley