to play devils advocate what you get when you use an out of the cms like drupal is someone elses logic. I personally find Drupal an absolute nightmare, the way i work is completely different and i can barely get it to work, thats not to say you will find that.
I get annoyed that certain things are extra plugins such as Keywords and description meta tags. Why are they not core! i just don't get it. I prefer to write my own.
As for it being more secure! well there is an arguement that many eyes have checked drupal but it is also true that it is easy for someone to find your site uses drupal from certain tags in the code and exploit any known security hole that comes up in drupal.
With a custom cms how does someone start hacking your site as they don't know what you have called your tables or even where your admin pages are. I know things can be modified but i don't like that user lists are not searchable and are just a long list of users. How can i find someone if the only thing i can do is sort by name and i have thousands of people. i'd have to click through 100's of pages of results manually.
Drupal also puts loads of extra code into the page with wrappers here there and everywhere. The entire point of CSS is to have nice clean code without pointless divs everywhere.
I often think of it in terms of cars. If you want a fast track car for £10k you can go buy a Ford fiesta and modify it a little. It won't be the best on the track but loads of mechanics can work on it and it'll be fairly reliable. If you spent that £10k on a custom built kitcar with sports suspension, spaceframe chassis, stripped down lightweight etc you will get something that is designed for the job it is doing. You might need a mechanic that knows a little more than the average joe but you'll get a faster outcome in the end
and call me cynical but a lot of website companies will use Drupal, wordpress etc as it is a massive time saver as they don't have to write their own cms and as you know time is money.