Do MPs need a degree or doctorate? No.
However, many jobs in the public sector require at least a bachelors degree, paid for by the person doing it, so spending 3 years, at least, to achieve the required qualifications with no guarantee of a job at the end of it.
Your argument about MPs is not really the full picture, as many will have had well paid jobs before entering politics. There are also career politicians, who have never held a 'normal' job. Many politicians will easily land jobs in industry once their nose is out of the Westminster trough, making use of contacts made during their time in office.
Politicians also get very generous pensions, expenses, subsidised food, etc... so not quite the sob story you're trying to paint. As far as I know, nobody else in the public sector gets all these extras, which equate to quite a substantial package beyond their basic pay.
Furthermore, last time I checked, to become a doctor or dentist takes a minimum of 5 years, self-funded study, having passed various stringent requirements to just get a place at university. For anyone wanting career progression, there are many more years of study required.
Ultimately, everyone makes their own choice as to what direction they take in the world of work. Be it self-employed, balancing risks and rewards, employee to somebody else, again balancing the costs and rewards. However, when people see those in power lecturing self-restraint, not just politicians, whilst ignoring the same advice and decisions dished out to others. It is this hypocrisy and lack of parity that riles people.