For the real answer on "kvar" we'll have to wait for a native to explain the nuances.
But let me give it a try... first let's get some trustworthy examples from svenska.se
https://svenska.se/tre/?sok=kvar&pz=2#!
EXEMPEL: use of kvar as "left behind"/"stayed"
- hon stannade kvar efter middagen; she stayed till after noon/lunch
- han lämnade kvar några böcker; he forgot some books
- hon glömde kvar sin kofta; she forgot her vest/coat.
- han stod kvar i samma ställning i flera minuter; he stood in the same position (he didn't move) for several minutes
The above examples don't apply to your case.
ABSTRAKT EXEMPEL: en avfolkningsbygd där det är svårt att hålla ungdomen kvar; hon vill till varje pris sitta kvar som ordförande
- sitta kvar : (wanted to) stay leader/president/spokesman
So "kvar" together with the preceding verb, expresses the continuity of some situation.
So now back to your sentence
"stannar kvar" expresses the continuity of being at home... not leaving (after all) for a holiday which they had in mind,
but wouldn't take place (maybe) because of the too many covid-related complications...