Is an all-road bike too heavy for climbing?

For most recreational riders, no. The typical 0.5–1 kg weight penalty compared to a pure road bike translates to roughly 5–10 seconds per kilometer of climbing at typical amateur power outputs. This is rarely significant outside of competition. Wider tires at lower pressures also improve traction on steep, loose climbs where a lighter bike with narrow tires might struggle for grip. If you are a competitive hill climb racer, choose a lightweight bike instead.