...Be6 might be one of those moves that tempts White into overthinking / overextending himself trying to find a crushing refutation where none exists.

In such cases sound, principled moves (such as your 8.Nf3/Nge2) may be the best "refutation."

I don't get many chances to play the Black side of a Closed Sicilian, or I'd give 7...Be6 a go  myself to see what happens.

Well, the very odd 7.Qc1 has been played by the anything but odd players Pachman and Smyslov - the latter beat Bronstein with it in the USSRch of 1951. Anyhow, after 7.Qc1 Ng4 8.Bd2 Be6 9.Nge2 Black's cannot play d5.

A few concrete lines mostly when white trys to win c5. There are some lines where white can win a pawn without losing but they really don't seem at all safe. It's possible that there is some quiet line which gives white an edge although I'm not sure why they should. Obviously I might well be missing some stuff:

6 Be3 Nf6 7 h3 Be6!?

8 Nge2 d5!? 9 Bxc5 d4; 9 ed Bxd5!? (a real computer move this) 10 NxB NxN and then 11 Bxc5 Qa5+ is bad so say c3/o-o Nxe3 etc. 

Alternatively 9 ed Nxd5 10 Bxc5 Nxc3 11 NxN BxN+ 12 PxB Bd5!? ^ Qa5 etc. (11 PxN Bd5!? 12 f3 etc is miserable & black can have his pawn back on c3 if desired.).

8 Nf3 d5!? 9 Bxc5 Qa5!? (9 .. d4 10 Nce2 Qa5+ 11 b4 etc doesn't work) 10 Bd4 de 11 de Nxd4 (Rd8!?,o-o-o!?) 12 Qxd4 Rd8 etc - this isn't utterly clear but fun.

... 9 ed Nxd5 and Bxc5 gets met by Nxc3 ^ Qa5 & 10 Nxd5 Bxd5 11 Bxc5 by Qa5+. So say c3 = ish.

8 Qd2 (critical perhaps) d5!? 9 Bxc5 

and now 9 .. d4 10 Nce2 and 9 .. Qa5 10 ed Nxd5 11 Nxd5 Qxc5 11 Nc7+ x e6 are no good. So black has to go for:

9 .. de!? 10 de Qxd2+ (Qa5!? is more logical I'd think) 11 Kxd2 Nd7 12 Be3 Nde5 is a fair bit of fun if mildly speculative. (Junior predictably loves it.).

10 Nxe4 Nxe4 11 Bxe4 and then Bxb2/Qd7 seem ok as does: 11 .. Bd5!? 12 o-o-o when it's just possible that the truly absurd looking 12 .. Bxa2!? works.
(certainly 13 b3 doesn't seem to work after Rc8. I still struggle to believe this one though 12 .. f5!?/Rc8/o-o etc are all reasonable and sane).

If white decides to put an end to this d5 nonsense with 8 Nd5!? things can still get rather wierd.
eg 8 .. Bxd5!? 9 PxB Nb4 10 c4 Qa5!? 11 Kf1 which is certainly pretty odd. Or 8 .. h5!? taking advantage's of white inability to play Nf3 etc.
Black can also just castle but it doesn't seem to be in the spirit of the thing somehow

It seems that it is probably possible to play like this after 7 Qc1 (7 .. Be6 ^ 8 .. d5) but white does get some extra Bh6 ideas cf 8 Nge2 d5 9 ed Nxd5 10 Bh6 o-o etc which seem to take a lot of the fun out it all.
Could try angling for a transposition via 7 .. Ng4 8 Bd2 Be6!? I suppose but I'm not sure if white will/should oblidge.

Isn't 7.Qc1 more precise? Or does Be6 work as good then?

Very odd, moving your bishop to g7 twice...