Your company pushes you to a managerial role because your experience allows you to deliver more value in that than you would as an individual contributor. You're are essentially a force multiplier for more junior ICs by providing the strategy, direction and mentorship to them. You'll find this career trajectory to be fairly normal.
Being a coder makes you a cost centre, whereas being a PM gives you some degree of charge over a budget, which means you can more directly quantify your $ contribution to the business. The upper end of your salary expectations will therefore likely be higher in the PM role. It sounds like you don't want to do PM, but sort of realise this.
There are a handful of exceptions to this, where you become such a domain expert that people will write blank cheques for you do still do the technical stuff, but you will find overwhelmingly that your experience level is one where companies see your best value in providing you the responsibility to guide the high level and leave the detail up to juniors below you.
Of course, you may choose that you want to become that greybeard IC, and that's totally ok, but expect it to put some degree of limit on your salary's upward trajectory. I don't think that they will kick you out just because you're old, but they certainly see you as being able to deliver more value in some degree of management. As an IC, you simply can't provide that much more value back to the company compared to juniors in similar roles, because your true value lies in being leverage to multiple juniors through your experience and mentorship. If you're paid well enough, satisfied with having that for the rest of your working life, then it's an increasingly stress-free way to pay the bills because your mastery of the role will mean it's easy to deliver.
It's worth noting, that some of the other ways around this (starting your own company, freelancing, consultancy) will inherently include a degree of management of some description. Whether it's the right balance for you is again something only you can gauge.
So if you truly want to stay as an IC, how do you do that? Talk to your supervisors about your career goals and highlight that. Have that discussion, explain to them that you're happy to take on a mentorship role but still want to be in the weeds at the end of the day, and hopefully your company is flexible enough to work with you there.
Context: I'm also 35 but in a mech engineer career, and in exactly the same career stage as you. I now have 3 people reporting to me, but more informally play a strategy/mentorship/leadership role to a team of about 10 more junior engineers.
I have thought a lot through this, and my team has multiple of those old greybeards who stayed in the technical expert IC role. I wrestled with the same issue (I used to love doing technical design, FEA, CAD, etc., and can still probably run rings around the juniors in it) but ultimately realised that I want the career growth and that I can deliver substantially more value by moving into a leadership position.