I think this is a good idea although realistically you can't get rid of expenses. The job means there will be loads of things where expenses will be incurred and you couldn't expect the employee to fund them out of their own pocket (whatever the salary they got). You could get rid of the system and require expenses to be covered by a credit/debit card that is issued to them - with strict rules on what can and can't be claimed, and harsh penalties for fraudulent use (like perhaps jail time for repeat offences).
As a formerly prolific business traveller I don't understand the issue with expenses - it's just people who don't know how that all works.
However the expenses should be, as they are for a business:
1. Paid only against receipts, other than small non-receiptable incidentals (though a fixed allowance is probably a better way to cover these).
2. Only paid where necessary for the business.
3. Not bringing any personal benefit or gain to the recipient (thus it's OK to claim rental of a property for a long term thing like being an MP, and the company has paid for a flat for me where it was cheaper than hotels in the past, but it's not OK to claim for a mortgage unless the company, not the employee, is purchasing the home for future employees to use).
4. Subject to the standard income tax related limits.
A corporate credit card could be used if it's felt preferable to pay-and-claim, I've had one of these in the past when I travelled often and it did make personal accounting easier.
Obviously, expenses fraud should and would be dealt with with due seriousness.
Likewise, having accommodation provided or an allowance would also make sense. Let people keep their existing properties (the role of MP isn't a job for life, so your own home shouldn't be factored in). If an MP has a long way to travel, give them a season ticket or another allowance for travel by car. Basically, do what is done for employees of private businesses up and down the country. I would have no issue with an MP having a season ticket and enjoying the benefit of free leisure travel as a result - as any season ticket holder would.
Indeed, this also makes sense. The one thing I'd say is that paying expenses to rent flats/for hotels is not practically avoidable, because a large dedicated MP accommodation block would be too risky a terrorist target. Having people distributed around different parts of London is much easier for security purposes and thus probably worth the extra cost. However as I note above, these expenses should never be for mortgages, only rental. A core tenet of business expenses is that the employee doesn't benefit personally from them, they simply cover their out of pocket costs to put them in the position they would be in if they hadn't had to travel.