Could you supply a web-adress to the dojo? then we could take a look and give you a better interpretation of the term 'traditional' used in the specific context.
On topic: Aikido differs A LOT. from very soft to very hard. From emphasis purely on technique and footwork to flow and dynamics and further on to the very selfdefence oriented - and all the different combinations in between. It usually comes down to 'the pedigree' of the dojo and - not the least - the the sensei in charge.
Aikido can be really really good for your overall health, but it is not a given. Some do pushsups and weightlifting to become physically strong. Some do ki-exercises for hours.
Regarding randori - it is usually only for the more advanced students, but we do 'limited randori' where you train the feeling of moving and taking initiative in the situation. It is good cardio training for both the person doing the waza and those attacking.
Aikido does however tend to be a little difficult in the beginning so it wont be a good workout until you get the basics down. Depending on ability, the people in the dojo and prior MA experience this might take anything between a few months up to years.
Best thing to do is to go observe a class - or maybe even join in - in what ever dojos that may be in your area, and then take a few days to mull it over. You should find a place that give you what you want, and where you enjoy the way it is given to you.
You might even find something that you didn't expect you would like. One of my students wanted to be like Steven Seagal and took little interest in the weapons work we do - but he's slowly warming up to it, and is actually thinking about buying an iaito for more serious sword practice. So stay open minded
Good luck with it