The unemployment allowance is for the temporarily unemployed. The question of whether the unemployable or those that have simply given up on looking for employment should be given a basic income is a different debate. Basic income is a debate that needs to be had, but simply massaging the unemployment allowance system we have and ignoring the real question is not the way to go.
No, the government should not pay the unemployed a monthly allowance without any question as to if they are seeking employment or not. There should not be free handouts for people who choose to be lazy and live off of others. However, there are reasons a person might not be seeking employment such as caring for children or elderly relatives. They may be pursuing other avenues to long-term employment that don't include an immediate job. These other factors should be taken into consideration because they are valuable.
The government should pay the unemployed a monthly allowance without any questions as to if they are seeking employment or not because the know the rate of unemployment is not Zero percent. So, until the rate drops to zero or to a lower rate, they should not questioned anybody in that case.
No, the government should not pay the unemployed a monthly allowance without any question as to if they are seeking employment. To do so would led to abuse, dependency, and the collapse of the economy. Historically, socialist countries who have made such transfer payments have not fared as well in terms of growth or production.