Should Americans show a vaccine card before they travel to another country?

That's up to the other country, but if I were in charge of passport control at a popular tourist destination, I would add it to the requirements.

Many countries require proof you've had specific vaccinations before they let you in.

That's up to the other country. Once you leave America our rules and laws don't apply.

Most countries that require travelers to show proof of vaccination are for mainly Yellow Fever, which is said to be fairly rare, and they only require that vaccine from travelers coming from an infected area. This means that it doesn't apply to the majority of first world travelers. These are nations in Sub-Saharan Africa and some parts of South America and aren't popular tourists destinations anyway. Most first world travelers do not actually need to show proof of any vaccination. Anyone telling you that Americans are required to show proof of vaccination has probably never left the country, are just flat out lying or are ignorant as to which countries require them and why.

That would be up to the other countries they hope to visit. 

Will other nations allow those Americans entry WITHOUT the vaccine card? I suspect it will be a matter of showing proof of vax or doing a 2 week observed (not honor system) quarantine.

Many countries have required proof of vaccinations in the past, so I don't know why this would be any different.  I'm going abroad in August, and assume I will have to prove I've been vaccinated. 

Those vaccine cards can be forged by the millions.  

Yes.  We're in a pandemic.

No, the very concept is absurd and vice versa, foreigners shouldn't have to show totalitarian identification to come to America either