The evidence strongly suggests that Alex isn't an Immortal
Around 21 minutes into the film, MacLeod indicates that he believed all other Immortals were dead, and that the Prize was his (this is shortly after he sees lightning streaking across the sky in Marrakesh, as a result of Kane beheading his minion, Senghi):
MACLEOD: How can this be happening? They were all dead, and the Prize was mine. So who is out there? And where is he?
Why would MacLeod believe there were no other Immortals remaining, if he thought Sarah Barrington was one of them, and knew of no evidence that she'd died?
Around 22 minutes into the film, MacLeod indicates that he needs to return to New York, the place of the Gathering, in order to find Kane.
MACLEOD: I know he's alive, and I have to find him. Going back to New York, to the place of the Gathering, is not what I had in mind. Once again, my life is about to change.
This suggests that MacLeod believed Kane would be irresistably drawn to the place of the Gathering, as all Immortals were supposed to be at the proper time, according to Ramirez in the first film:
RAMIREZ: You must learn to conceal your special gift, and harness your power, until the time of the Gathering.
MACLEOD: What Gathering?
RAMIREZ: When only a few of us are left, we will feel an irresistible pull towards a faraway land... to fight for the Prize.
And New York is precisely where both Kane and his other minion, Khabul, went, after leaving Nakano's cave in Japan, with no other in-story explanation given for that, other than the implication that they were drawn there by the Gathering. Which begs the question, why wasn't Alex drawn there, if she too were an Immortal? She did eventually go to New York, but only as part of her work. There was no suggestion that she was irresistably drawn there.
Around 39 minutes into the film, Alex translates one of the inscriptions found on the walls of Nakano's cave as the phrase:
ALEX: There can be only one.
She then asks herself:
ALEX: Only one what...?
This clearly suggests that Alex had no knowledge of the Game at this point in the story. Which begs the question, why not? And how could a female Immortal have managed to survive hundreds of years without knowing about the Game, without being on holy ground, and without having any apparent means of defending herself? She should've been easy pickings for other Immortals, if we go with your theory.
Around 44 minutes into the film, Kane is able to approach her in her New York office without being detected until he announces himself. If she were an Immortal, why didn't she sense him? Also, why didn't he take her head, or show any awareness of her being an Immortal?
Around 47 minutes into the film, Alex is able to tail MacLeod in her car, from his home in New York to the Japanese Spiritual Centre in Queens, and subsequently approach him on foot, with MacLeod reacting as if he were unaware of her presence until she spoke to him. If she were an Immortal, why didn't he sense her? The Immortal sense is clearly a thing in this film, as MacLeod repeatedly senses the approach of Kane and/or his minions during the film, like when Kane shows up in the same building just a few minutes later.
Also, if MacLeod thought Alex was Sarah, why does he treat her as if she's someone he's just met in this scene, and subsequent ones? He shouldn't feel any need to maintain the Russell Nash pretence with a fellow Immortal who was also a former lover of his. And if he didn't think Alex and Sarah were the same person, then why do you question his judgment in that regard? Wouldn't he be a better judge than either of us?
Around 67 minutes into the film, Alex approaches MacLeod in the Scottish Highlands, with him again showing no awareness of her presence until she speaks to him. The subsequent dialogue they exchange is pretty telling as well:
ALEX: You seem like you belong here.
MACLEOD Maybe I used to... but that was a long time ago.
ALEX: You spend alot of time alone.
MACLEOD: I'm used to it.
ALEX: Nobody is. You don't let people in, do you?
MACLEOD: I can't. It hurts to love and...
ALEX: ... and then see them die? Like the woman in the painting?
MACLEOD: Yes. I loved her very much.
Those last couple of lines strongly suggest that Sarah Barrington was mortal, and that both Alex and MacLeod viewed her as a separate individual, rather than a younger version of Alex. This contradicts your theory that Alex was regaining memories of a previous life as Sarah. The dialogue which immediately follows that above contradicts it even more:
ALEX: I know who you are. You're Connor MacLeod. Born into the Clan MacLeod, in Glenfinnan, on the shores of Loch Shiel. You were driven from your home in 1536...
MACLEOD: ... and I cannot die...
ALEX: ... and have wandered the world ever since.
If Alex were merely regaining memories of a previous life as Sarah Barrington here, then she wouldn't know anything about MacLeod being an Immortal, or being driven from his village in Glenfinnan. Sarah was never shown to be aware of any of that. Alex was clearly piecing together the clues uncovered by the investigative work she'd done on MacLeod (since that piece of Scottish tartan was discovered in Nakano's cave, much earlier in the film), along with everything she'd witnessed since then, including hearing Kane address 'Russell Nash' as MacLeod in the Japanese Spiritual Centre, seeing Kane transform into a bird and fly through a window, and seeing photos on the wall of MacLeod's home, where supposed ancestors of his had his exact likeness.
MacLeod basically admitted he was hundreds of years old to her when he confessed to having loved Sarah, a woman who lived during the time of the French Revolution.
Around 80 minutes into the film, Kane reiterates the idea that MacLeod, himself, and his two minions were the only Immortals remaining after the events of the first film:
KANE: Oh, I know, you must've been very disappointed. All those nice years, thinking you were the only one left. But you see the Prize was never yours, and now, it never will be. Amen.
Why would Kane (correctly) deduce that MacLeod had been under that impression, if he (Kane) believed Alex was an Immortal too?
Lastly, let's consider the final words spoken in the film:
MACLEOD: There's a fresh breeze blowing off the loch. The glen is snowy and cold; it's the most glorious sight on Earth. I've found no place to match it in 400 years. I'm back now, with a woman I love, and a son I've taken as my own. I'm at peace; it's finally over.
This strongly suggests that the Game is over at this point; that no other Immortals remain, and that there's no more fighting to be done. If Alex were truly another Immortal, MacLeod shouldn't be at peace here, because it'd mean the Game wasn't over. Remember, one of the central tenets of the franchise is the phrase:
In the end, there can be only one.
MacLeod himself says this repeatedly, in both the first film and this one. Virtuous Immortals such as MacLeod and Ramirez may be able to resist killing one another whilst other Immortals remain, but the logic of the Game has always implied that the last two remaining Immortals must fight to the death, even if they're friends, or lovers. If we go with your interpretation, the ending of this film is completely at odds with the whole premise of the Game.
As for the flashbacks with Sarah being pointless if she wasn't Alex, I disagree. Firstly, flashbacks of some kind are almost mandatory for a Highlander film, especially this one, which is clearly doing it's best to copy the formula of the original. (The producers of the Highlander franchise knew they screwed up badly with all that planet Zeist stuff in Highlander 2, so they played it ultra conservative with Highlander 3 by retreading the beats of the first film so closely, as to make it all but a straight remake.)
Secondly, those specific flashbacks with Sarah establish that there was at least one other major love of MacLeod's life -- aside from Heather -- prior to him establishing his Russell Nash identity in New York, which helps to fill out his backstory.
Thirdly, the emotional vulnerability that MacLeod shows when he confesses the love he felt for Sarah to Alex sets up the romance between him and Alex, by confirming to her that he is in fact a romantic; something that wasn't all that evident from everything she'd seen of him up to that point.
I will grant you that the striking resemblance between Alex and Sarah was a curious red herring in the film, if we assume it was purely coincidental (and no alternative explanation was explicitly given, or even implied, really). However, as unsatisfying an explanation as that may be, it's much more plausible, and far less convoluted, than assuming that they were the same person, considering all the apparent plot holes and contradictions which go along with that theory. The principle of Occam's razor is certainly not on your side here.
