Thanks for posting your question.
I understand your frustration as to the lengthy wait to get an approval for a rental. In an ideal situation, the landlord would promptly assess your application or at the very least, keep in communication with you regarding a status update of some sort. In a situation like this, I'd suggest allowing the landlord their 'end of the week' time frame request and if you still have not heard from them by that following Monday, then just kindly reach out to them again to check the status of the process and request a response as to where they're at with your application.
I truly hope you get the place, because application fees are not refundable and after a while it can get very expensive to continue to apply for more rentals.
If this doesn't work out, I would highly recommend you to find a local realtor who is willing to help you find a rental (rentals pay little commission, but some realtors might be willing to do it in hopes of earning your business and/or referrals in the future) and make sure to ask the landlord or property manager PRIOR to applying:
a) will they process one application at the time or do they pick the best application out of a pool of applicants? ;
b) how long will it take to get an approval/rejection from them.
Hope this information was helpful!
Wish you success, :)
Coldwell Banker Residential RE