League-of-legends – Why buy doran’s starter items

Same reason people buy pots, including the refillable + variants. They give a better early boost than anything else. Often the dorans are meant to give a similar level of sustain, but with more stats. Dorans ring allows you to use more than 10 spells before level 6 in the mid lane. Dorans shield (which doesn't actually get used that much due to blade / ring / corrupting being better on the current meta tops, and tbh in general) allows you to withstand some harass and heal it up while still csing. Dorans blade can give even more healing than pots if you play safely as you should if you need that healing. They also give a wider variety of stats usually than what else you could start with, you COULD start with a ruby and get 180 health + some pots, or long sword for +10AD + 1 more pot even, or you could get a dorans blade and get 70hp, 7ad, and 3% lifesteal which not only is factually more expensive stats that you can't get any other way early, but can also give even more healing. While yes this doesn't save you the 350 gold or 400 gold build into an item that you want, it allows you to stay in lane and not only be the same level (and therefore able to trade) as your opponent, you also will (hopefully) not miss so much cs that you're even more down than that 450 gold.

You usually don't sell this until you're buying your 2nd / 3rd item and that last bit of gold puts you over or until you need more space in your inventory to finish something more useful, so you make a very good use of those stat boosts by using them as long as you do.

Occasionally in dominant lanes top I'll buy long sword + 3 potions or refillable instead. I often do this on trundle, because his passive gives so much healing that you don't need more on attacks usually, and his q is such a good dueler of other melee champions that even with 10 ad vs 70hp + 7ad + 3% lifesteal (significantly better / more expensive stats, especially early) I can still dominate trades. A lot of people also go corrupting on trundle, I find the long sword is more damage, the normal potions are more healing, and he just doesn't need the mana unless you're wasting his Q on cs. Of course I'm only a 2x ex plat and current gold so I'm definitely not the end all be all of what's the best start on champions, but I know enough about the game to make informed decisions (and hopefully help you do the same), regardless of whether it's best or not haha.

I'll take Ben's rep here too by saying what he did as a tl;dr: "They are actually some of the most gold efficient items in the game, give lots of good stats at a good price, give you better sustain and trading than other starting items, and are priced so you can still get potion(s) at the start of the game."