I-joists on an old dimensional lumber floor
I think an I-joist is a good solution.
Your existing 2x10 floor joists have dried out over time and are stable enough to mix with I-joists.
However, I question the 7” allowable hole size. I’d double check with your I-joist manufacturer and follow their guidelines for size AND distance from bearing point.
There are many sizes and grades of I-joists, so be sure to coordinate all that with the manufacturer. I have no doubt the 9 1/2” I-joists will be strong enough for your 13’ span.
I’ll give you a third option: Option 3: Use the 9 1/2” I-joists at 12” on center. Align the lower chord with your existing 2x10’s (this allows you to use the existing bearing walls, beams, etc.) so no adjustment for replacing the ceiling. Then, use 1/2” plywood subfloor instead of replacing your 3/4” subfloor. (1/2” plywood on 12” centers is as “stiff” as 3/4” plywood on 16” on center.) You won’t notice the difference.
The underlayment can be reinstalled or the existing underlayment or it can be removed throughout the space and new installed for uniformity.