This is where the crux of all LCD slow-start problems lies. Capacitors are tricky little things when you're dealing with technology like a television that constantly suffers changes in temperature. Fundamentally, the hot-cold nature can cause several problems in the capacitor's ability to retain charge without destroying the rest of the board.
A little physics/electricity review is in order. In case you've forgotten (or this is new to you), capacitors in general are simple electrical devices that are all about storing a charge once your computer has been turned on. The electricity that the computer needs often needs to be redundant. In the event that electricity is lost, these capacitors can retain enough charge to "finish the job" in terms of keeping your computer's essential parts alive enough to have a good shutdown. As a fun little aside, if you've ever had a desktop, once you turn off the PC, you'll be able to hear the capacitors losing their charge once the electricity goes out - this will be a high-pitched noise that comes a few seconds after total shutdown.
If a capacitor in a non-pivotal area of the main board of a TV goes down or becomes inefficient, you'll get slower startup times as components cannot get their life-giving electricity in the same amount of time that they used to. This will lead to a crippling slow-down with startup time.
Remember - even though the slow startup time can be a pain to deal with, fixing it will more often than not be too much trouble to actually deal with. It'll cost a lot of money and time to try to fix a problem that essentially is only applicable a few times a day. If the monitor is taking a few minutes instead of seconds to boot up, a new monitor can run you less than $100 nowadays for top-of-the-line technology as long as there is a deal going on.