This explanation might answer some of your questions MaizieD
An independent and neutral head of state
Even if an elected head of state has previously been a party politician they can still be independent and neutral (impartial) once in the job. The rules of the job would require them to be non-partisan and because their actions are open to scrutiny the public and politicians can judge whether those rules are being followed.
Britain already has plenty of examples of people in these kinds of positions, most notably the Speaker of the House of Commons. The Speaker is elected to parliament as a party MP, but once he or she is chosen as Speaker they remove themselves from party political debates and instead represent the whole of the Commons and act as referee in MPs' debates.
Candidates may be people with successful careers in law, business, foreign affairs, teaching, science, or someone who has made a name for themselves championing a popular cause or running a big charity. In a great country like ours, with over sixty million people, we'll be spoilt for choice.
The experience of other countries
Other countries have heads of state similar to the one Republic supports for Britain. Most notably is our nearest neighbour the Republic of Ireland. Ireland has elected a series of excellent presidents including Mary McAleese (so popular she was re-elected unopposed), Mary Robinson who went on to serve with the United Nations and Michael D Higgins.
Germany's head of state is elected by an assembly made up of national and regional politicians. In recent years two presidents have had to resign for either breaking the rules of the job or getting into political controversies. These resignations and the way in which the presidents were replaced show how robust the democratic alternative is. When an elected head of state gets it wrong they are held to account. If a royal gets it wrong it's either laughed off or excused without debate.
The Italian head of state has been central to holding the state together while Italy suffers political and economic crises. This is the role of referee in action, as the politicians argue about forming a new government the head of state can keep things together and provide continuity. In Britain if we faced a similar crisis we would have no-one but our MPs to look to, the Queen simply cannot play the same role as arbiter and referee (and made a point of refusing to get involved after the 2010 election when we were left with a hung parliament).
Not like the US or France: A very British head of state
Republic supports a non-partisan head of state who is not involved in making political decisions or running the government. So we don't support a system like they have in France or the United States. We believe the best alternative to the monarchy is a head of state who is able to do the job that the Queen cannot do. It is a serious job of representing the nation, acting as referee in the political process, championing the interests of the people and defending our democratic traditions.