A higher profile leader leading to a more successful Free French was certainly possible, but depended entirely on events during the last days before French Capitulation.
When the government abandoned Paris and relocated to Bordeaux, the righting was well and truly on the wall; France would fall. Premier Reynaud proposed to General Weygand that the French should do as the Dutch had done; seek a military capitulation only, while leaving the government freedom of action to continue the struggle from overseas. Weygand indignantly refused and Reynaud, despite having the authority to dismiss Weygand for failure to carry out orders, did not do so, instead he resigned; he was by that stage on the verge of a mental and physical breakdown. If Reynaud had dismissed Weygand and appointed someone else as commander of the French army, with authorisation to capitulate only those forces in mainland France, those members of the government willing to continue the fight could have flown to either North Africa or England and been able to issue orders to the French colonies with the full authority of the legitimate French Government.
The French cabinet were divided between those in favour of capitulation and those wanting to continue the struggle from France’s overseas empire. In Anglo-American terms these would have been ‘the hawks’ and ‘the doves’, in France they were referred to as ‘the hards’ and ‘the softs’, which possibly suggests something concerning French though processes at the time. By the 14th of June, the softs formed a definite majority of the cabinet, but there were still key members in the hards camp, principal amongst them being Interior Minister Georges Mandel and Defence Minister (and former Prime Minister) Edouard Daladier, both of whom headed for North Africa to try to continue the struggle from there. The Minister for the Navy, Cesar Campinchi and the Minister of State, Louis Marin, were also leading hards.
If either Mandel or Daladier (or in a best case scenario, both) had flown to England but otherwise events continued as IOTL; Reynaud resigning, Petain becoming premier and asking for an armistice, then they would have had much more pull than de Gaulle had, both in terms of being able to persuade French soldiers there to join the Free French and in politically when dealing with the British and Americans. Daladier in particular, could have used the dissolution of the French parliament on July 10th and Petain’s assumption of the presidency to denounce him and declared that he, as most senior remaining member of a legitimate French cabinet sworn in by President Lebrun, was assuming the duties of the premier and leading a French Provisional Government in Exile.
As heads of an exile government, neither Daladier nor Mandel would have been perfect, but then an obscure Brigadier-General with no political experience and a prickly personality wasn't either. Daladier was a virtual manic depressive who sustained himself through the crisis of 1940 by drinking heavily; many witnesses describe him reeking of Absinthe during cabinet meetings, but his public image was stronger and inspired more respect than his private one. Mandel’s character was outstanding for the role of an exiled leader; he’s was calm, unflappable, resolute and determined, the only thing going against him was that he was a Jew, and this was something that the Vichy regime and the Germans would have used for maximum propaganda effect. The best scenario would have been for both Daladier and Mandel to have got to England, preferably with Campinchi as well; Daladier would have become the head of the provisional government with Mandel providing the reliable anchor and capable administration, while Campinchi would have been able to issue orders to the French Fleet with at least some confidence of outweighing the orders issued by Darlan.
None would have equalled the prestige of Petain, but they all had more political prestige, clout and public profile than de Gaulle had, with the possibility of more colonies and men joining the Free French cause.