Thomas Rogers' wife Alice

Are you referring to Alice "Elsgen" Rogers

It appears her baptismal name was Alice. I don't know how the name "Elsgen" became associated with her but it more seems to be a historical nickname.

I wouldn't change it without documents that show otherwise. There are more docs becoming available on line through the Leyden archives

George J. Homs is this something you could take a look at?

Tough one. The profile for Alice "Elsgen" Rogers suggests hat she died in Leiden in 1662. That looks very unlikely. In any case, there is no death record on that specific date in the Leiden archives. I'm seeing that various authors suggest that Thomas Rogers married twice. I would think that Alice is indeed not Elsgen. Though a transcriber may have interpreted Alice phonetically as Elsgen - Elsgen is typically a short name for Elisabeth. This would indeed suggest two marriages.
Whatever I try on the site of Jeremy Bangs, I don't find much for Thomas Rogers. It looks the only source seems to be a poll tax record. All that makes sense, given that the children of Thomas Rogers all seem born in England. If none of them married in Leiden, an indeed soon left Leiden to follow their father's traces - extremely unlikely that any records remain.
My suggestion is to follow what various authors seem to say: two wives for Thomas Rogers - one Alice Cosford, another Elisabeth "Elsgen".
A death for Alice Cosford in 1662 in Leiden seems totally unlikely... Her children left for America. I would think her only reason for staying in Leiden would be that she married there for a second time - but then their should be traces for that. Plus, that doesn't explain that Thomas Rogers also was associated with a wife named Elisabeth. Also, I don't see traces of a Cosford (or a variant) in Leiden.

Is there something that verifies that the children left for America? Also, if Elsgen is the short name for Elizabeth, what is Lysbeth the short name for?

Don't know if it helps but I found this: Per author Eugene Stratton, the 1622 Leiden poll tax listed the family of Thomas Rogers residing there in poverty, but apparently without his wife Alice. Instead of Alice, this family included a possible second wife named Elizabeth – or the Dutch variant Elsgen, and with the children of Thomas Rogers – son John and daughters Lysbeth (Dutch for Elizabeth) and Grietgen (Dutch for Margaret).[2][6]

So the information is agreeing.

1) the wife of Thomas Rogers appears to have died before the 1622 poll tax. Her name was Alice and many have her maiden name as Cosford.

2) there was an Elizabeth living in the Rogers household in that poll tax of 1622; a variant name for Elizabeth in Dutch is Elsgen, another variant name is Lysabeth.

I updated the death date & name for Alice.

The next step is to determine if we should be making a 2nd wife profile & what details if any ...