Although there is no definitive answer, I found this Reddit post that states that the BDSP and PLA starters giveaway in HOME were shiny-locked, despite the Galar starters being available shiny. The link inside the post no longer works, but OP states this has to do with the upgrade to HOME 2.0. This would lead me to believe that the Paldea starters and any other event giveaways are shiny-locked as well.
I also found this other Reddit comment that states "Only the galar starters all the other starters you can get from home are shiny locked."
I could also find no evidence online of someone receiving a shiny Rotom or other starter giveaway from HOME, while there were plenty of posts describing encounters with shiny Galar starters in HOME.
