Sounds very much like you've both got a failed film capacitor across the output (usually an orange philips 0.68uf). Replace this with a 0.22uf (250v or 400v) and it should sort the problem.
Although the motor is actually 0.39uf at 50hz/33rpm, it's better with 0.22uf as the cap needs to be half the value at 45rpm (67hz!). Therefore you'll be ok on 45, but 'a bit out' on 33, but no major problem so it will start on either speed.
These decks are all 20-25 years old now, but the fact that it runs on one speed means the amplifier and generator are WORKING - just not properly on 45. As the original cap is much too high for either speed (+ aged as well), it will just stall on 67hz which is not surprising really.