This is just peoples way of scaring others away from property investment, which personally is good for those of us that know the truth. The reality of getting a late call is not all that common, but I can't say it never happens. The clogged toilet, does happen, but guess what, that is why you hire a local handy man and get to know him/her. Cause for $15-20 they take care of this little stuff and you never deal with it. This is also why you background screen your tenants and how you choose your investment properties. There is a science and an art to picking the types of properties that attract the right people.
As for a REIT over direct property, REIT's are not bad but they don't compare to direct investments and as @aliston point out you miss out on appreciation. Overall from what I have seen a decent REIT might get you 10-13% annually, which isn't bad. However, it is quite a different type of investment that is also subject to the stock market turns, interest rate changes and other financial market forces which have negative changes more often then does the raw property investment. Hence, unless you have substantial amount to invest, getting much out of a REIT will take a while.
As for rentals, they are not a get rich quick scheme, but over relatively short times they make you comfortable and give you security in passive income. Over the longer term the property values along with some rehab projects can make you "rich" and then if you keep at it, mixing in some commercial property is always good, you have the possibility of becoming wealthy.