[CLI] Speed up `sky status` 40% by reducing the health check

[Michaelvll]

This speed up CLI operation on remote API server from (reducing RTT from 14 to 9):

Note, the following is with a remote API server and no managed jobs or services available

The RTT reduction leads to 0.9s time reduction

The concurrency leads to 0.45s time reduction

Local Master

===> multitime results
1: sky status -u
            Mean        Std.Dev.    Min         Median      Max
real        3.486       0.142       3.335       3.453       3.747       
user        0.590       0.028       0.551       0.594       0.630       
sys         0.124       0.015       0.109       0.116       0.150       

Local PR

===> multitime results
1: sky status -u
            Mean        Std.Dev.    Min         Median      Max
real        2.071       0.038       2.011       2.075       2.130       
user        0.550       0.020       0.527       0.546       0.584       
sys         0.108       0.007       0.098       0.108       0.119   

Disclaimer: this PR is mostly done by Claude 4-sonnet

Tested (run the relevant ones):

• [ ] Code formatting: install pre-commit (auto-check on commit) or bash format.sh
• [ ] Any manual or new tests for this PR (please specify below)
• [ ] All smoke tests: /smoke-test (CI) or pytest tests/test_smoke.py (local)
• [ ] Relevant individual tests: /smoke-test -k test_name (CI) or pytest tests/test_smoke.py::test_name (local)
• [ ] Backward compatibility: /quicktest-core (CI) or pytest tests/smoke_tests/test_backward_compat.py (local)