I have a good career (I don't know if I'd consider it 'high flying' as I'm an academic, and I'm not sure any of us are 'high flying' anymore). But I'm well educated with a very fancy sounding title working for one of the more prestigious UK universities. I only have two children, but I don't think it's impacted my career, and while I have no desire to have a third and never have, the only difference I can see would be childcare costs.
School run/sports days/school plays are largely the same time out of your day if you have 1 or 3. Childcare will cost you more, but if you have the means to pay, in theory, it shouldn't make any difference if you are putting 2 through nursery or 3. You still have to be off work to collect them at the same time.
We also don't have any family help and all sick days/holidays are between the two of us. Honestly, I don't find it a massive imposition. Dh and I share holidays and use a holiday club and I don't personally run out of AL ever. Mine have rarely been sick for any length of time. Perhaps I'm more hard nosed than most, but they always go into school unless they have a fever or D&V (or like chicken pox/COVID). No one stays home for a cough or the sniffles or a duvet day or because they didn't sleep well the night before. We had COVID over Christmas last year (so all off anyway), and other than that, they each may have had 1 or 2 days off the whole year. We both support each others careers and no one carries all the load. When I'm busy, dh is doing the school run at 3pm and homework all afternoon, and when dh is busy, it's me. We work flexibly and always have, even before dc.
Mostly, I think anything is possible if you can buy in the help. But really, with just TTC #2, see how you get on and then make a decision. Having more than one adds a whole extra level of complication and challenges that have nothing to do with career pressures. It's two sets of emotions. Two bath and bedtimes to do (so you get very little downtime in the evenings). Two rounds of night time or early morning wake-ups. Managing two sets of friend groups. The fighting. The fighting is horrendous. 😂You may find you are satisfied with stopping at two once you have them.