How to outmatch Eldritch Blast...

On the paper, Eldritch Blast with Agonizing Blast is king... But it's overdone and boring. Surely there's a better way? What about Shadow Blade? That's 4d8 per hit, and you get advantage when not in Bright Light. Here's the problem: it takes a bonus action to cast it, and another bonus action to make it your pact weapon and use your spellcasting modifier and get your extra attacks. That's the entire first not being able to attack with it. Plus, it requires concentration, so you can't stack it with Spirit Shroud.
On the other hand, Eldritch Blast with Spirit Shroud feels like a better option, AND give you the extra benefits of Spirit Shroud, AND will be effective on your first turn.
You could also take Magic Initiate (Druid) from Lessons of the First Ones and get Shillelagh on a Club.
Let's do some math with average damage, assuming we're always hitting. In tier 4 play, assuming 20 CHA, you get for each case:

With 2 dual-wielded Shillelaghed Club +1 (assuming your DM let's you keep your weapons in that state), you will deal 13 damage + 9 from Spirit Shroud, for a total of 22 on hit, and 17 with the off-hand. With 3 attacks + 1 off-hand, that's 66 damage total on the first turn (you need the BA to cast Spirit Shroud), and then 83.

Now, Shadow Blade deals 23 damage on hit. With 3 attacks, that's 69 total damage on average per turn. And that's only from round 2.
But if you have a Shillelaghed Club +1 as a Pact weapon before casting Shadow Blade, your first turn will still deal 39 (or 42 if you cast EB with AB) damage, and from round 3, you will get a 4th attack that will bring your total to 77.

With Eldritch Blast (+Agonizing Blast), you get 10.5 damage + 9 from Spirit Shroud, for a total of 19.5, but you get 4 beams, for a total of 78 average damage per turn, from round 1.

So Shadow Blade is not the best option, unless you take into account the guaranteed advantage in Dim Light and Darkness that Shadow Blade gives you. Then, depending on the enemy's AC and your spell attack modifier, Shadow Blade might be better because you'll hit more often.

But then, what if you have something better than 2 Clubs +1? Let's say you have 2 Vicious Clubs instead.
With 2 dual-wielded Shillelaghed Vicious Clubs (again, assuming your DM let's you keep your weapons in that state), you will deal 19 damage + 9 from Spirit Shroud, for a total of 28 on hit, and 23 with the off-hand. With 3 attacks + 1 off-hand, that's 84 damage total on the first turn (you need the BA to cast Spirit Shroud), and then 107. NOW it seems like Eldritch Blast sucks.

But having 2 Vicious Clubs isn't very realistic... What if we only get 1, what can we do with our bonus action? How about we take a Lessons of the First Ones and add the new Spellfire Spark Origin Feat, which let's us cast Sacred Flame as a bonus action...
Our first turn with the Pact Shillelaghed Vicious Club + Spirit Shroud still deals 84 damage on average, and the next one deals 102. Still better than EB alone!

But what if you use Eldritch Blast + Spirit Shroud AND a Vicious Club as offhand?
You'd need Spell Sniper to use it in melee, but on the first round, you still get 78 damage, and then 99. Pretty darn close, but still less than the dual-wielded Vicious Clubs option. 

And the Shadow Blade setup with a single Shillelaghed Vicious Club will deal 57 damage on the first turn, then 69, and then 83, which is still far behind.

So Eldritch Blast is still excellent, but can be outdamaged if you use Shillelagh with a Club. Shadow Blade just can't keep up, but in a dark environment, having advantage on the 3 main hand attacks might still make it a preferable choice...

And then there's Blade of Disaster, which deals 70 damage per turn as a BA and makes Shadow Blade and Spirit Shroud obsolete.
Without Spirit Shroud up, EB only deals 42 damage per turn on average, while the Shillelaghed Pact Vicious Club will deal 57. No contest :)
If all you have is a Club +1 however, then your damage with it will only be 39, which won't be enough to outdamage EB+AB, but you'll still get a +1 on attack rolls and a better hitting chance.

Why not just go pact of the blade GWM?

EB+AB = 
1-4th = 1d10+3(4).  8.5, 9.5 at 4th
5-11th = 2d10+8 (10).  19, 21 at 8th
12th-17th = 3d10+15.  31.5

Pact of the blade w/GWM at 4th: (LVL 1 Ability scores Str 13, Dex 16, Con 12, Int 8, Wis 8, Cha 17)
1-4th 2d6+3 = 10, 12 at 4th
5-11th 4d6+6+6 = 26, 28 at 8th, 30 at 9th
12th-20th  12d6+15+12 = 48, 51 at 13th, 54 at 17th

I like the idea. But then you have to invest in STR. That means you'll have to sacrifice your other stats, and replace one feat.
A Pact Vicious Maul with GWM and a PB of 6 deals 4d6 + 11, that's 25 damage on average per hit and 75 per turn. Neat! And no need for Shillelagh.
And if you add Spirit Shroud, that's 102 damage on average... Yeah, that's indeed really nice :)
I was suggesting pure Warlock, but with this strategy, you could justify dipping in Fighter to get GWF and the Topple mastery as well.
GWF will make your 4d6 deal 16 on average instead of 14, so that's an extra 6 damage on average per turn (and the assurance that you won't roll low).

My first comment is that, if EB is overdone and boring, don't play it. (Yes, I know you're theorycrafting. I'm just saying.)

Next, Shillelagh can't be used on two weapons at all, and only lasts a minute anyway, so your arguments against shadow blade apply also to it.

EB+spirit shroud is an extremely short-range option, so you're going to need dex and con for AC and HP.

And if you need dex anyway, then you can attack with shadow blade even without making it your pact weapon. True strike plus AB (assuming your GM allows it, which is way more likely than double Shillelagh) plus bonus action attack with dex is probably worth running the numbers on. 

Also, evaluating based on how it plays at max (or at least 17th) level distorts things.  Bladepact's third attack comes online at level 11 (12?). There's a lot of game between that and your 4th EB, and more damage now is likely more valuable than best damage later.

Yeah I forgot you couldn't double Shillelagh, but Spellfire Spark solves that issue (or simply dual wielding with something else).
True Strike is 3d6 at level 17. With Shadow Blade and AB, that's 18 + 10.5 + 5 = 33.5 damage per turn on average. Not bad, but still lower than everything else that's been suggested.
Elfdope's suggestion is actually much more effective, especially if you dip in Fighter to get a weapon mastery.
Another thing that deals a shitton of damage every turn is Danse Macabre, but you need the corpses.

Also, I'm surprised you didn't mention this, but I've just realized that I can't use EB and then use my BA to make an off-hand attack. You need to take the Attack action with a Light weapon first before you can make an attack as a BA with your off-hand weapon.

Quote from Natrel >>

Yeah I forgot you couldn't double Shillelagh, but Spellfire Spark solves that issue (or simply dual wielding with something else).
True Strike is 3d6 at level 17. With Shadow Blade and AB, that's 18 + 10.5 + 5 = 33.5 damage per turn on average. Not bad, but still lower than everything else that's been suggested.
Elfdope's suggestion is actually much more effective, especially if you dip in Fighter to get a weapon mastery.
Another thing that deals a shitton of damage every turn is Danse Macabre, but you need the corpses.

Also, I'm surprised you didn't mention this, but I've just realized that I can't use EB and then use my BA to make an off-hand attack. You need to take the Attack action with a Light weapon first before you can make an attack as a BA with your off-hand weapon.

 pure warlock? celestial patron, vicious greatsword, agonizing blast, spirit shroud (radiant) green flame blade, reppeling blast, and you've got 10d6+3d8+15 damage. that's alot.

I'd go Fiend Patron instead of Celestial so you can use Hurl Through Hell and deal an additional 8d10 (avg. 44) psychic damage on your third hit.
Lifedrinker, GWF and GWM will also increase your damage.
And if you use Blade of Disaster instead of Spirit Shroud, the damage is just insane:
20d6 + 12d6 + 15 + 18 + 8d10 + 1d6 = avg. 70 + 48 + 15 + 18 + 44 + 3.5 = 199.5 avg. damage per turn.