Well, here are 5 reasons why:

1. It will give you the ability to 1v1 anyone in the early game and almost everyone at any point in the game and be victorious as long as you have it available.

2. You already have a "flash" ability wich is, of course, your Repel. Use it to engage, disengage and follow the enemy.

3. You will build Mobility Boots and Rylai's Scepter wich will give you move speed advantage (combinded with your spider form MS bonus) and Rylai will slow the enemy so much that even if they flash you won't have any problem catching them (yeah, you can finally beat up Vayne even if you miss the stun)

4. You also have your stun ability when in human form wich is more than enough to make sure your enemy has no escape

5. No one will expect you to have ignite as Elise, even if they see it on the loading screen they will, for most of the time forget about it because everyone is used with Elise to play with flash, so use this surprise element in your advantage

I've been playing with flash since I started playing Elise and most of the times I forgot it was there. It's just not worth it on Elise. I would prefer to have the confidence that I can kill anyone that invades me in the jungle (wich happens alot when you play Elise) than being afraid that I will not be able to kill that Lee Sin or Lee. Oh, by the way picking Ignite also automatically counters Master Yi for example and renders their heals useless. There are just too many advantages to not go for this composition every single game.

It makes you have so much trust in yourself when you're playing Elise.
It secures you so many kills and it makes your life a whole lot easier.
It's just a better option than anything else.
Combine that with the Smite and..see what happens :)

Remember, do not Ignite the enemy the second you see them. Instead, ignite them after you land a stun and/or engage them and they are at least close to half hp, otherwise you will waste the ignite for nothing for most of the time. They will heal and come back to get you; And you have no flash so this is no joke, be very careful of when you use your ignite, only use it to secure your jungle and other 1v1 engagements and kills.
When you have a Master Yi in jungle coming for you then yes, you can use Ignite earlier in order to make sure that his heal will not be that effective but you gotta make sure you land a stun on him in order to get the kill.

It's like a new meta that was always there in a way but strangely I haven't seen anyone using this strategy, even the best Elise players still use the flash so I hope people will realize how much better this is for Elise.

In the early game you will want to focus on clearing your camps as effective and as faster as you can. Go for ganks ONLY if you can secure the kill, meaning only if the enemy you're ganking is overpushng their lane.

When engaging a lane in early game DO NOT USE THE HUMAN FORM STUN FIRST, use your Q ability in order to scare the enemy laner and make them flash, if they don't flash you will use the stun since you had enough time to get close to them by now, use the W and if you get them with the stun quickly Smie (if available) and Ignite (if available) and go in to the spider form then use Q and W and repeat unil they're dead.
If you can afford to not use the Ignite while ganking (ex: your teammates assist you) DO NOT USE THE IGNITE. As long as the kill is secured do no waste your only spell that can help you defeat the enemy jungler in a 1v1 jungle encounter afterwards.

Go for first blood if possible at level 3, then go for mobi as soon as you can. You don't need to complete the skirmisher sabre in early game. The mobility is a much better choice since you can clear the jungle and gank faster and it can also help you evade ganks. If you don't want to go back to base two times (first at 900 gold) and second at 650 gold in order to build the skirmisher as well then stay jungle or try to get a kill on one of the lanes in order to reach 1500 gold then go base and you will be able to get both items while also saving precious time.

The moment you have mobi and skirmisher and ignite available you can basically gank whatever lane needs help and 90% of the time secure the kill for your laner or for yourself.

The next item you'd want to get after these two will be the upgraded Skirmisher's Sabre (Runic Echoes, the blue dot one). That will make your ganking and your 1v1 dueling abiility even more easier and will also help you clear the jungle monsters faster.

You will want to go for the first Dragon the moment you hit level 6 and/or you have the Skirmisher's Sabre finished. After you upgrade the Skirmisher to Runic Echoes go for the Rift Herald. When at the Herald hit him so that you wake him up and get back towards the wall. This will give you the option to repel onto the other side in case you get ganked while doing the Herald. After you go in the spider form position yourself towards the tail while the herald is busy attacking the spiders so that when the pink sign appears you can hit it and so you can defeat the herald faster.
Always fight the herald near the wall since you have no flash in case you get ganked and your repel will only work if you have something to repel onto.

Go for the Rylai as soon as you finish the Runic Echoes upgrade. You desperately need that slow when engaging, especially if the enemy ADC is someone like Vayne that can easily doge your E stun. The slow will make sure you stand a chance towards that type of enemies even if they are ahead, since you have the ignite as well.

After you take out these two objectives pay attention to the the score tab, see wich enemies are ahead in the lanes and gank there. Always help those who need help on the lane, do not waste your time helping lanes that already do good. Buy vision wards whenever you can and deward whenever you can using your Oracle Lens.

Roaming and Ganking is essential during the mid game with Elise. This is the phase in wich you are at your best so use it to your advantage and always keep the ignite for when you really need it.

I will keep experimenting with the runes and come back with what would be a better option, for now just use DH or Electrocute.

I am experimenting with a combination of Pecision and Domination at the moment.