According to NoA's support website, access to Pokemon Home accounts is based upon the Nintendo account associated with the console (emphasis mine):
Pokémon HOME on Nintendo Switch is tied to your Nintendo Account, so you can connect with any Nintendo Switch console that your Nintendo Account is associated to. For example, if your Nintendo Account is linked to two different Nintendo Switch consoles, you can use your Nintendo Account on either console to connect to your Pokémon HOME account.
This means that if you sent Pokemon from the broken Switch to Pokemon Home, you would be able to access those Pokemon on a different Switch if it used the same Nintendo account as the broken Switch. The first Switch being broken does not change this because your Nintendo account exists on Nintendo's servers, not any console.
However, the answer to the question in the title is no, because the above would not allow you to access Pokemon from the save file of the broken Switch. As save files are stored in the system memory (source), you would need to access the system memory again to retrieve the Pokemon. (Pokemon in Home are not in the system memory.)
The best way to recover the Pokemon would be to contact Nintendo for repair of the console. If you do this, then you should be specific about the fact you're interested in the save data first, as Nintendo are notorious for clearing system memory during the most innocuous repairs.