Laura as a Landlord you should be acting professionally and should be displaying due diligence regardless, there should be no emotion as you described as this is business and business should not be emotional i.e. based on feeling, gut, face value etc.
Please note from experience my words that things are never an issue until they become an issue and with property when you have an issue it can be financially crippling for the landlord.
BTW i have had a house in London empty for well over 2yrs now with -£0 rent income and a loads of outgoings for instance and still no light at the end of the tunnel.
I shall never again accept anything less than 3 months bank and payslips and credit checks and ll reference & employer reference and passport, however i usually obtain much much more data in reality like social media checks etc
Please also remember as Colin states that a payslip is easy to make/obtain/falsify hence it should only imply they have a salary as it is the bank statement which is needed to a-company the payslip in order to validate it and further as Colin covers i agree and expand that you need to remember that you have a property worth how much? = £500,000 for example yet if you had the same amount in cash in your bag would you hand that over to the tenant too i wonder, i think not.
i thinks some Landlords, more-so the inexperienced ones don’t really comprehend fully that they have something which is worth a lot, hence it needs protecting a lot as when your tenants burn it down or invite bed bugs/termites and your gets stripped down to the brickwork and un-rented for years as i have had before then it can be fully understood yet better you lear from our mistakes in advance and be smarter and wiser quicker.