Based on the specific conditions in the apex controller, you can setup same conditions using validation rules on Lead object. This will work both in SF Classic and Lightning (then no need for the visualforce page).

E.g validation rule for the first condition on Promo__c field

IsConverted && ISBLANK(Promo__c)

Create other validation rules for conditions on Qualification_Score__c and Flag_Type__c

Make sure that Require Validation for Converted Leads is enabled under Lead settings.

Based on the specific conditions in the apex controller, you can achieve same conditions using validation rules on Lead object. This will work both in SF Classic and Lightning via the std button and lead path (so no need for the visualforce page).

E.g validation rule for the first condition on Promo__c field

IsConverted && ISBLANK(Promo__c)

Create other validation rules for conditions on Qualification_Score__c and Flag_Type__c

Make sure that Require Validation for Converted Leads is enabled under Lead settings.

The validation rule will fire after the user clicks on Convert on the lead conversion page.