2013 MacBook Pro Logic Board

One of them told me the risk was that the logic board for a 2013 would be "obsolete" by 2021.

I have no idea what that provider was talking about either. Apple published repair/service policy† says in most jurisdictions, they will provide service and parts support for five (5) years after date of last production. After that you have to find a private provider, if they can get the parts.

For an Early 2103 15" MBP, that date was Oct 2013 so its parts/service support ended in Sept 2018. The Late 2013 15-inch MBP date of last production was July 2014 so its parts/service support ran out in June 2019.

If the provider is talking about OS support running out in 2021, his crystal ball must be a lot cleaner than mine!

† — Ref: Vintage and obsolete products - Apple Support

When owners of the 2011 model started developing failures of the Discrete graphics chip, a number of work-arounds were developed by hobbyists. These essentially turned OFF the discrete GPU and/or made it invisible. The Integrated Graphics chip then did everything, perhaps slightly slower.

But there is no pathway to an External display from the Integrated graphics chip, so that feature will be lost by applying any of these work-arounds.

Apple has tightened the requirements of posting work-arounds here, to protect the innocent. You have to deliberately ASK to have them posted, and acknowledge that they are not supported by Apple and could conceivably cause damage to your computer (but there are no reports of damage to date).

Is this the 15-in model with dual Graphics?

do you need an external display?