Do starting characters begin play with 0 Reknown?

Yes. And they won't like it

Edit: No one forces you to give your players only 0 reknown. If you want to start at a higher enemy power level for example you should give them some reknown. For example we had one session in the beginning where the KT had to fight against banshees with 0 reknown. It ended in a lot of destroyed chain swords due to enemy power swords...

What do you think of players starting with 1 or 2 to represent the reason they were chosen to join the Deathwatch?

As far as I understood renown was earned as a member of the Deathwatch not as a member of their parent chapter.

Anyone heading for the Deathwatch is already a renowned member of their own chapter, chosen to represent them.

I could never understand why the Tyranid war veteran should get an extra lot of renown that bleeds over into the Deathwatch

You begin your tour in the Deathwatch as equals with zero renown. Starting with 1 or 2 simply because you were chosen for secundment means EVERY Deathwatch marine would have 1 to 2 renown, which completely defeats the purpose. So no, you start with ZERO.

The only way to get more renown at start to my knowledge is to be a Tyranic war vet and even then it's up to 1d10 worth

You begin your tour in the Deathwatch as equals with zero renown. Starting with 1 or 2 simply because you were chosen for secundment means EVERY Deathwatch marine would have 1 to 2 renown, which completely defeats the purpose. So no, you start with ZERO.

1-2 is next to irrelevant anyway.

By default and by RAW, everyone starts with Renown 0.*

As a GM, you can choose otherwise as you please, but do understand that it makes more (and arguably better) equipment available to the PC, as well as decrease one of the build-in motivators for stories.

* barring oddities like Tyranic War Vets and possibly some stuff from RoB