Then yes. Especially with a guitarist that has effects. Sometimes you will need to add an octave or two up to play a clean rhythm guitar part to keep the song musical if he gets too into the effects. solos, time & key changes, etc.
You will want to have the option of a chorus, tape delay. At least I would in that situation.
You should have a compressor in this situation. Just like your EQ you can't really play live without it.
You may be able to rely on the amp for any dirt you want, or not have any dirt in your sound. Both are acceptable, but a dirt pedal also gives you the option to turn it on and off quickly without going to the amp settings. If you rely on your amp for dirt you are adjusting at least two knobs usually to keep unity volume and reduce the dirt.
Rocky Mountain Way(godsmack)
Breed
Shine
Open Up Your Eyes
Comedown
Machine Head
My Own Worst Enemy
All The Small Things
Pour Some Sugar on Me
Paralyzer
Aint Talkin Bout Love
Eruption (You Really Got Me)
So both Bush songs, you should try playing with a kind of spitty, uneven fuzz. You may not like it but that's generally how they are played. I'd say Nirvana's Breed that wouldn't be a bad choice for it either. You may not like it for the whole song, only certain phrases. Generally Bush songs sound better with distorted bass of some kind. I've seen them play Machinehead live with just Gavin and an acoustic guitar. So there's that.
Sugar give it a try with a bass chorus. You might hate it. But for a 3 piece playing it you are going to lose some of the sound the rhythm guitar would add without either chorus, octave, or both. Personally I'd go with both.
Both Van Halen songs don't need any but wouldn't hurt if you added something. Try a flange (one of the cornerstones of the EVH guitar sound, may or may not compliment how your guitarist plays them) or phase, and as well some chorus, delay or both can add some dimension to both songs that otherwise are very treble/guitar heavy.
Here's what you should do if you are going to get some pedals. Look at Joyo, see what I describe above and what others suggest, and decide what's worth the money to you. If you like whatever you get from them, either keep it or get a nicer version of that effect. You can always turn off and/or keep off some effect that doesn't sound right in the room or in the song.