No. As I said the estrogen receptor is activated differently according to what is binding it. This is true for both ER-alpha and beta. I wrote a review paper on this some years back, for graduate school not publication: maybe I can find it for you (but it may not be on this computer and if not then it is not practical to find. It certainly isn't on this computer as any directly-existing file but may be buried within something else as part of some kind of backup. Just don't know at the moment.)
I know of no evidence that this is the case for the androgen receptor. (Btw when I use that construction that is not a smarmy way of saying "I know there is no evidence," it is just a literal statement that I don't know it -- but it's not that I know all the evidence in this field.)
And as I said with the SERMs, a given one can be an agonist in one tissue and an antagonist in another. (Which differs from your statement regarding regulation of the number of receptors.) If anyone has evidence that this is ever the case for the "SARM"s that would be very interesting. I haven't seen it. So far as I know they are nothing but agonists.
As for the claims of no side effects: There are plenty of women that have been prescribed, for example, Anadrol at 50 mg/day with no side effects either, except on the liver, which has nothing to do with androgenic properties but rather the alkylation. And plenty with oral Primobolan at that dose and no side effects at all. And back when the synthetics were first developed they were fairly commonly given to women and children. It is EASY to set up a study in such a way that a measurable therapeutic outcome will occur but quite likely no adverse side effects will be reportable. I am not at all sure anything has been demonstrated with regard to the "SARM"s in this regard as has not already similarly been demonstrated in many studies with plain ol' anabolic/androgenic steroids.
(Note, please no one interpret the above as meaning those doses carry no risk for women. They do. The point is that medically they were considered non-problematic, based on clinical studies and even in practice. Primobolan has never been available in the US, but Anadrol was used for quite some time. For that matter Winstrol is probably still used for women but at quite low dose, but still enough to claim therapeutic benefit. So that it can be claimed that the "SARM"s have "no side effects" does not prove, merely from the claim, that actually anything different must be going on.)