I'm guessing that this isn't a newish TV?
There were a number of flatscreen TVs made as crossover models; PC monitor / TV. This is going back a while, late 2000s to early 2010s IIRC. I supplied a few for various jobs. The difference compared to a regular TV was the resolution and pixel response timing.
A small TV would be no better than 1366 x 768 pixel resolution which is classed as HD ready. The thing is, at the time, that the average desktop or laptop didn't have a default standard* output mode for that. The result could be that the TV would try to scale the PC image, and that would then give fuzzy text and odd picture artefacts. These PC Monitor / TV products had a screen resolution that a computer would be happy to drive without having to jump through hoops running some custom scaling software. This image wouldn't need the TV to scale it. Overscan (a TV feature) would be switched off as well which further helped with PC display.
The pixel response was faster too. TVs at the time didn't need that, but your computer did. The pixel grey-to-grey times were the same as for standard PC monitors. This helped prevent ghost trails from the mouse pointer and other screen issues.
Anyway, lots of TV screens of the time used CCFL backlighting. This is cold cathode fluorescent lamps; miniature tubes. The power supplies for those were invertors, and the power board contained capacitors which, at the time, had a bit of a reputation for going bad.
The early signs were exactly what you're seeing now. The TV would be difficult to start. It would be fine though once the caps warmed up. Eventually though the caps wouldn't hold sufficient charge so the TV backlight would die.
The solution was to replace the caps. You could go mad and replace all of them, but that's a bit OTT. Finding the one(s) that need replacing though does require a bit of skill. Sometimes you can spot where the top has started to bulge or dome. Other times it might be the seal at the bottom where the cap sits on the board that has started to squeeze out. Other times, especially with intermittent caps, there's no clear visible sign, so you have to then use a different technique.
Capacitors respond to heat and cold. An intermittent cap can be provoked to go faulty if chilled. (Use freeze spray directed at one cap at a time.) They'll come back to life when gently heated. (Use a hair dryer with the air blast directed at a single cap if possible.) When a TV is failing to start on the first power button prod, heat up the caps first, then try a 'from cold' start where it would normally fail. If the TV comes on then there's a good chance you know it's a cap and you can continue narrowing down the exact culprit.
Replace with the same values for voltage and capacitance. If possible, upgrade the temperature rating from 85 deg C to 105, and choose a low ESR version. This means the capacitor itself is designed to stay cooler during its charge/discharge cycles, and that prolongs the life. It also reduces ripple current which gives the components further down the line an easier life.
You can get more info and pictures of what to look for here:
https://www.badcaps.net/forum/forumdisplay.php?s=981b818bfe897f93a54f657e86de74d0&f=24
'* Yes, PC graphics cards can do that now, and yes, with scaling software back then it was possible to have all sorts of custom resolutions, and yes (again) with a high-end PC graphics card there were more options yadda yadda yadda. What I'm talking about here is the average spod's PC where they were running a fairly basic graphics card and just wanted it to work.
