I have a very similar Meddings machine, and very good it is too.
You are right – your machine has been damaged. The motor-mounting must be all true and square for two reasons.
– Firstly so the pulleys and belts align correctly.
– Secondly, I think your drill has a different motor-mounting to mine, but if yours also has a spring belt-tensioning arrangement, so it works correctly. (Mine has a warning-plate that the belt tension must not exceed that given by the spring.)
I'd dismantle the back end, and examine each component to determine which one(s) is / are damaged. The bars might be bent on their spigots in the plate. If the plate is mild-steel, that might be bent. It would not be too difficult to make replacements, in fact easier and more certain than trying to straighten the existing parts.
It could have been dropped but I would expect other damage to be apparent. Alternatively, it had been lifted badly, putting excessive strain on the motor mounting.
That original plate looks upside-down too – hence that adaptor between it and the motor. It's possible the set-up there is not helping by giving an extra cantilever effect.
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I've occasionally used mine as a tapping-head. Electricity off at the mains, belt slipped off, and turning the spindle by hand, using the chuck. The work held slightly loosely laterally to give a self-centring effect. One hand rotating the chuck, with a leather glove for protection from the sharp edges on the chuck; the other gently easing the quill down and up.