If one has been always paying Medicare taxes (withheld by employers), then Medicare hospital plan (part A) is premium free providing acute hospitalizations. Employer Group plans depending on plans provide longer hospitalizations, but they have premiums to pay.
Medicare medical care insurance (part B) pays for doctor visits, medical tests etc. It has monthly premiums, yearly deductible and Medicare don’t pay whole amount of covered services; remained amount is small coinsurance to be paid by insured. One has to calculate costs and benefits for comparison. Medicare coverage may give better benefits than basic low cost employer plans, but it also costs more than simplest employer base plan with high coinsurance. Especially medical tests can be very expensive if not covered by employer coverage.
Medicare advantage plan (part C) benefits are similar to employer group plans, because both use HMOs, PPOs, and several others. Benefits can be better than those of standard Medicare for example preventive services like exercise programs, prescription drugs and eyeglasses, which standard Medicare does not cover. One still has to pay Medicare premiums and health care organizations may charge more. Usually employer pays part of premiums like 50%. So employer coverage usually costs less than Medicare advantage coverage.
Depending on private insurance benefits and individual’s health it might be good to have employer coverage for benefits and Medicare coverage for future, also because if not applying for Medicare when eligible premiums go up. Since one’s health is unpredictable, people with higher incomes buy Medicare supplemental policy to help paying future health costs. People with lower income typically use Medicaid as supplemental Medicare insurance.