Although the oud can play microtonal intervals, does Middle-Eastern music still usually use scales that are made up of 7 diatonic notes (such as the Phrygian), that can be played on "non-microtonal" instruments such as the piano? In other words, are the scales that are used in the Middle East usually just a "mode" of Western scales?
No. Often, they are not modes of Western scales. On my algerian mandole, I really do need the 1/4 tone frets to play some of the mini scales that (when combined) produce a maqam.
It really gives a distinctive (uncomparable to western scales) colour to the sound. Although some of these mini scales perfectly fit the western Major or Minor tri/tetra chords... others (which are my prefered) just do not fit at all.
On my instrument there are only two 1/4 tone frets, and I think it the case for all algerian mandole. So it is nowhere as versatile as an Aoud you are refering to, but it is far sufficient to give access to a bunch of those micro tonal mini scales, which are NOT AVAILABLE for a standard western guitar fretted neck.
I don't remember the names of those musical forms. I have to search through my notes/links. I will update this answer accordingly with relevant names and information.
You are really asking two questions here: "Are scales in middle eastern music usually 7 notes?" "Can most middle eastern music be played typical on Western instruments?"
The answer to the first question is "kind of." You can conceptualize most (though not all) Arabic music in terms of 7-note scales if you want.
It's a pretty rough appoximation though.
It's not how the music actually works and ends up being misleading. The longer answer requires understanding that "scales" in the Western sense are not really a generative or organizing principle in Arabic music. The underlying principle is the jins (pl. ajnas), as noted by Stephane in his comments. These are smaller, localized groups of notes that are combined to create maqam music. Most music uses multiple ajnas starting on different notes throughout the piece, in ways that can't be described in a simple scale concept. The closest we come to this in western music would be the way the melodic minor scale is used.
The answer to the second question is categorically "no" - the two most common maqamat in middle eastern music are the Rast family (which has microtonal 3rd and 7th degrees at least) and the Bayati/Ussak family (which has microtonal 2nd and 6th degrees at least). Very few pieces of middle eastern music avoid microtonality entirely. Even in pieces that more or less correspond to a mode that we have in Western music (Nahawand/Buselik = Minor, Kurd = Phyrgian, Ajam = major), there are usually sections that are in a microtonal jins. But there are definitely some pieces which are possible to play without microtones, such as Lamma Bada Yatathanna, or Bint al Shalabiyya.
The violin family, being fretless, is certainly capable of playing these scales (it is popular in middle eastern music), but most Western violinists cannot do it (unless they have studied middle eastern music). Other instruments can do it to some extent with practice: trumpet, clarinet, saxophone all have flexible pitch and a musician with good ears who works on it can do it.
A good resource on this is the website www.maqamworld.com. One of the authors of that site has an excellent (and peer-reviewed) book "Inside Arabic Music" if you really want a deeper exploration of the music theory.