I 've done a few tests on this very topic and IMO the 34 is not as good in all the facets you mention.
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With a speaker that inefficient you will need a single-ended amplifier of much greater power- 50-100 watts, that is if you don't want to drive the amp too hard. Single-ended amps do their best if not driven past about 20-25% of full power.
OTOH a push-pull EL34 amp will likely work just fine. Generally speaking, you get a greater percentage of usable power out of push-pull amps although they tend to have less low level detail than a single ended amp. However push-pull almost always have more bandwidth (SE amps are limited by their output transformers), usually in the bass since most speakers that are efficient enough to work with SE amplifiers tend to not cover the bottom octave.
You can get around the low level detail issue by simply making sure that the P-P amp you get is not too powerful (the reason being that most P-P amps have a certain amount of minimum distortion which happens around 2-5% or so of full power; at power levels lower than that the distortion is actually higher). So if you are thinking of a single pair of EL34s in push pull you should be fine.