Not a City or PL fan here, but I can definitely see the reasoning for his signing. Stay with me here: (skip to the tl;dr at the bottom for a conclusion)
As a neutral I was really looking forward to seeing Pep deploy Sané on the right & closer to the goal. The position where he thrived in for Schalke, destroyed Madrid in the CL, & we saw his natural goal-scoring ability thrive. Rocket of a left-foot & had a wonderful way of beating his man, then opening his body to cut-in for a shot on goal from the inside the box. Always unpredictable with his movement. That Sané is easily a 15 goal a season winger when he comes to grips with the PL, at least I thought so before Pep started playing him on the left & saw him in a different role for the better of the team.
On the left he's of course still as capable of winning his 1v1's with ease, & sprinting towards the touchkine for a quality cross. He's arguably more predictable now there though & I'm not sure you guys are really making of the best of his abilities imo. Like I said before, with the way he was developing st Schalke, he really was on course to be the inverted winger with a Robben-esque playstyle capable of scoring 15 goals a game on the right. That sort of goal threat is just an added dimension to the team, makes defenders even more wary of the direct wing play, & is something that is much needed in the attack.
Goals are the name of the game & especially so in a front three. Finishing & compsure in the box has definitely been a problem for you guys this season, & Pep has said the same thing. There were many times where there was the potential to kill games off completely (many times teams came back to draw, or ended up winning due to it), or routing opponents 5-0, but just weren't capable of doing so due to the problems finishing the steak dinners in the box from all those amazing chances KDB & Silva created. Statistically, you guys had the best attack in the league according to most expected goals models, & underperformed all season.
To combat that, I'd love to see something like this.
Alexis/Jesus - Agüero/Jesus - Sané/Sterling
Sterling can play on the left obviously, but I think on the right is where Pep sees his development long term. He also plays at his best on the right, & has played there all his career, but I digress.
So assuming Pep starts to develop Sanè as an inverted winger more, that front three of Alexis, Agüero/Jesus, Sané in particular just screams goals & unpredictability for defenses to me. It's way more in tune with the type of front three template that Pep's had traditionally in Barca & Bayern (when Robbery were fit). Now that you guys are signing Mendy & Walker (hopefully), there's now that crucial support from dynamic & proper FB's as well. The wingers now have more viability to play narrow & closer to goal than before since they don't have to be the main providers of width & be in charge of the dominant supply of crosses. That's obviously one place where City lacked last season, it's crazy how many issues good FB's solve in the modern game. This also gives Pep even more flexibility formation wise, but that's that's a different story altogether.
Last season attacks down the wings became predictable in a way (although they were still effective), & add to that the fact that many times last season only one of Agüero & Jeus were available. Suddenly it's just one recognized & reliable goal scorer in the team the opposing team has to worry about. Neither KDB, Silva nor Sterling are that accurate in front of goal (although KDB's goal threat is definitely the best of the three) & both KDB & Silva have a propensity for shots from outside of the box from poor angles. Not to mention Sterling's way more of a threat beating his man on his strong side when isolated, winning penalties & causing sheer mayhem down the right where he'll inevitably try to find space for a good low cross to Agüero or Jesus. His forte has never really been in his composure in front of goal, although it's clear he has improved in that aspect this season. Definitely the Coman-esque player of the team.
So I can see why Pep would be interested in Alexis to help solve that issue. He's positionally versatile within the front three, & there'd be healthy rotation/competition on the left & at CF. It assures the players are at their best at all times, & provides key rest during congested fixtures. In game, no longer would there be one strict template on both wings. Relying on either Sterling or the most predictable & one-note winger in earth in Navas down the right, or just Sané on the left. Instead there'd be very different winger archetypes Pep could deploy & that in turn keeps opposing managers/team on their feet regarding to what sort of attack they'd be facing. Playing against Robbery is completely different to playing vs Coman & Costa (individual quality aside). Pep would have the option to play perhaps with more narrow full-backs & more traditional, wide wingers playing on their dominant side like he did many times last season. Or with modern, dynamic FB's marauding down the flanks, leading to a narrow front three closer to goal or perhaps instead opting for more midfield control.
Now it's only one Agüero injury away from relying way too much on Jesus for goals primarily during a congested fixture run-in, & with all the games this season, it'd be tough for him to be 100% for every game imo. Even more so if Sané exclusively plays on the left with more defensive duties & Sterling on the right. Alexis solves many of those issues, even more so if on top of that we see Sané gradually playing on the right & with more freedom closer to goal.
Only problem with Alexis is his national duty & personality. There really are no friendlies for Chile as each match, no matter the importance, the players put it all on the line & go full-out either way. So the CONFED cup this summer is serous business, & I imagine Chile will go very far in that, at least semifinals. The last four WC qualifiers for Chile (take place during two breaks, two matches each, 8/31 & 9/5, then 10/5 & 10/10) are must wins as there's no security for qualification in the top 5 apart from Brazil & Uruguay. Then the 2018 WC is the following summer. That is a lot of football in the time span with little breaks. Plus, Alexis absolutely despises being on the bench, he'll want to play every minute possible if he's fit. At least every CL & PL match, sure he won't mind the odd rest for EFL or early on FA cup matches.
There definitely is that worry about running himself to the ground considering his extremely energetic & intense play style. As the way he plays & the constant pressing he does for Chile & partly Arsenal is already up to par with Pep will likely ask of him whether it be on the wings or at CF. I'm not sure how sustainable his output will be at this level though once he hits his 30's, since the guy just never stops running & always puts his body on the line, particularly when he plays for Chile. I think there will a lot of nail-biting each international break & these next two summers, just like Arsenal fans.
(tl;dr) Whether Alexis comes or not, I hope Pep starts playing Sané on the right wing & closer to goal (like how he was developing during his Schalke days). As that added goal threat is needed on the right, & it's also a much more dynamic attack similar to Pep's front three templates during his Barca & Bayern days (when Robbery were fit). I think Sané could easily be a 15 goal a year player if he isn't burdened with as many defensive duties & can cut with his dominant foot on the right wing.
Also, I believe Alexis would be a brilliant signing & possibly even needed. Provides healthy competition & crucial rotation options. Though there are worries about his output/fitness once he hits his 30's due to his playstyle & his commitments with Chile. Very curious to see how he adapts his game once he's 31/32. Hopefully it's a Ronaldo-esque approach, & not the Rooney way...
