Working as a back bench MP was historically a part-time job, and would have to pay more to provide solely for the lifestyle of someone so important. As the answer by Michael Kay to this question explains:
Historically, MPs were usually well-off landowners; and being an MP
was a part-time job. They continued to run their estates and draw
rents from their tenants while travelling up to London occasionally
when required.
This changed gradually, but there was never a single point in time
when you could say this was no longer acceptable. There are still some
MPs today who fit into this tradition, and it's hard to argue that it
was OK in the past but is no longer OK today.
The practice of lords and wealth families in the U.K. having both a London home and a home somewhere in the country from which they received farm rents, is also an ancient one in the U.K., which is historically attested for centuries.
While this is less true in the U.K. parliament than it is in the U.S. Congress or state legislatures in geographically large states in the U.S., it is still true the many MPs have to maintain two homes, one in the capital in or near London, and the other in the locality from which they are elected. So MPs are under more pressure to earn money, and their pay as MPs doesn't go as far as people who only have one home.
Allowing them to publicly work for pay makes them less susceptible to bribery from non-employes as they seek to maintain a lifestyle suitable for what they perceive as their station.
Relatedly, one can afford to have more MPs for the same budget, at the rate of pay necessary for an MP to relying on their public salary as their sole source of income, than one can if one pays MPs at the current rate which can be supplemented from other sources.
These arguments are somewhat undermined by what MPs are paid, however, which is a fairly comfortable salary that addresses the two residences and travel expense issue:
The basic annual salary for an MP from 1 April 2024 is £91,346.
MPs also receive expenses to cover the costs of running an office,
employing staff, having somewhere to live in London or their
constituency, and travelling between Parliament and their
constituency.
The modern trend, however, is for the job to demand more time from MPs which puts pressure on their work-life balance. As one anonymous person explained:
Certainly from my own experience of working for an MP, whilst
Parliament is sitting an MP can work around 70 hours per week,
sometimes more. During recess the working week is much the same - at
least six days, with trying to keep much of Sunday for themselves
(although occasional constituency engagements may happen), so the
week's working hours may be more like 40-50. When Parliament is
sitting, MPs usually travel to London on the Sunday afternoon or
evening (many will return to constituencies on the Thursday evening),
so it's uncommon that there is ever a completely free day in the week
(holidays excepted, although work is still taken and MPs will still
keep in contact with their office and colleagues).
Marc Carter in the same thread states:
I doubt there are many MPs who work less than 35 hours a week.
Allowing MPs to work at second jobs when MPs are truly working full-time as MPs is more problematic, because the more hours an MP works as an MP, the more the pay that they receive for their good paying full-time jobs appear to be pay for who they know, and not for what they do, stinking of corruption.
The U.S. Case Compared
Notably, most U.S. state legislatures also have part-time legislators (this is also true in Switzerland). And, part-time legislators in U.S. states are often paid far less than a full-time employee at a mid-level professional position.
The practice in that context is justified, in part, because "citizen-legislators" are believed to be more in touch with the realities of every day life faced by ordinary citizens than full time legislators.
Many U.S. states also praise the part-time legislator model for the express reason that it reduces the amount of time spent legislating, which in turn, may reduce the amount of unnecessary legislation that is passed.
But, again, these justifications fall apart when supposedly "part-time" legislators are working 2000 or more hours a year.