When you create a card you get a personalised message containing all the relevant location hashtags and pet preferences as hashtags e.g. #catsitting or #reptilesitting . People who have those interests (pet owners, travel-affine) would then see it more likely (besides your own connections who might know someone etc.).
Another idea would be to share the card on an Instagram profile and run targeted ads to wherever you want to sit next - so pet owners from there can reach out to you directly. And if there is no luck through direct connections, there are still the good old local facebook groups :).
Especially on a place like LinkedIn however which otherwise is full of "I have a new job"-posts, pet content serves as a good antidote - and LinkedIn's algorithms are very much in favour of showing content to 2nd and 3rd connections. I think LinkedIn is a good place to experiment since there are many professionals in stable life situations who do travel abroad once in a while and maybe even create lasting personal connections across countries.
In general it is more meant to break the ice, get the word out, leave an impression and maybe next time your 1st, 2nd or 3rd connections need a sitter they think of you.
I actually had the backend built, with user access, password etc so users would have a live card on the page they could share - but the larger the project grew, the longer changes took - and I did not like the idea of becoming a middleman myself, be in charge of the data and user account / database management. It would have become yet another platform.
This was really the result of recycling the idea and re-empathising the M in MVP and making it more accessible for people to spread the message :)