The requirements for single family houses should be clearly explained in whatever code has been adopted by your town or state. Unfortunately, such codes require lumber for any load bearing framing member to be stress graded (with grade stamp on the member) so I suspect you are out of luck unless you can successfully claim the building is exempt for historic reasons.

Your recourse in such a situation is usually a "construction alternative" which, in most jurisdictions, must be prepared and stamped by an engineer or an architect. Otherwise, the inspector's opinion is final unless you want to waste a lot of money in a fruitless appeal. You are fortunate that the inspector was willing to suggest a solution since that is your responsibility, not his. You can't object to free design services.

I don't know what is supported by the 2x6's, the spacing or the span but if it is an attic floor, they must support 20 lbs/s.f. live and 10 dead load or more and be continuous from top plate to top plate in order to tie the exterior walls together (assuming the roof above is pitched).

For No. 2 or better Spruce/Pine/Fir lumber the maximum span would be about 11 ft. and less if the spacing is greater. The existing framing has some strength but an assessment of what that actually is would need to be done by a design professional if the inspector questioned it's strength. I'll warn you that few design professionals will make definitive statements about such issues.

If you were doing structural work without a permit, the inspector is being very lenient to not order the work stopped and a set of design drawings submitted.

Hmmm, I'm not a professional, just a homeowner. So FWIW, what I know about it is that if you start moving/removing walls, you are usually into "permit" territory. When you have new electrical work, and/or new plumbing (versus replacement), you are into "permit" territory. So, going forward be conscious how much you "tear into". We have hired a structural engineer for consultation and to give us his official assessment of what is needed more than once. I think it cost around $200-$300 for that. It is well worth it and usually will make the building inspector happy (at least in our town).

Adding a 6x10 beam requires a permit in any jurisdiction that has a building code. In fact, such a beam is probably too large to be prescribed in the header table in a residential code so it would usually be required to be designed by an engineer (a local lumberyard can often supply that service if you use an LVL beam). It is also necessary to prove to the building inspector that the new point loads at the ends of he new beam are carried down to a suitable foundation.

Any system that is disturbed in a renovation is usually required to be brought up to the new code for construction.

You can be sure the inspector's visit was not accidental; someone called him so don't imagine he'll let anything slide on this project. He'll want the permit fee too.

We have a permit, but the inspector was no accident. The electricians called him. We will probably have to lump it and do the job, but I can't help but think that the inspector is a bit picky. Thanks for the replies.

"workers cottage"

This reminds me of an acquaintance who bought a three storey semi-attached house built in the 1870s-1880s. He started renovations by removing the kitchen flooring. The more layers he removed, the greater the slope. Investigating further, he found it was originally a porch built on three-foot wooden piers, which had been enclosed and to which two floors had been added. The rest of the house was similarly jerry-built from scraps.

When I visited him, I opened the front door and the entire house was nothing but framing from the roof to the basement. He was in a corner of the basement digging out soil for a new foundation. By the time he was finished, the only original elements of the house were parts of the exterior walls.

Good luck!

There is a common misconception about code.
You should understand that it is the LEGAL MINIMUM
ALLOWED BY LAW.
That said, there is a LOT of room for improvement
over the minimum safety.
Have a conversation with your inspector, like
Mac said..its an opportunity, and yes, he was
very lenient with you.
He could have shut your job down.

best of luck

"...building structures frequently fail after many decades of use without noticeable sagging."

But very few fail period.

While codes have gotten much tighter over the years, few of the changes have to do with flat out failure of the structure.

Things like joist spans have been reduced over the years as materials dropped in quality, and flex in joists caused drywall (and sometimes plaster) to crack.

The joist spans based on 1/360 deflection are for the finished surface below and have nothing to do with actual structural integrity.

There remain thousands of old houses with foundations that would not be acceptable today (like un-mortared stone blocks weighing 300-400 pounds each).

I have seen many of them suffer failures from modern alterations that removed their support.

Diggning right beside a stone block foundation without adding permanent shoring for support against lateral shifting to put in a new addition is a recipe for disaster.

Houses do collapse without warning. I had to take a course on how buildings fail for continuing education credit. Many collapses are spectacular and result in lawsuits.

Here is a link that might be useful: house collapse

Here is another one. Oddly, the public has little interest in such stories. Even though many people resent the imposition of building departments on how they can build on their property, they appear to have no fear whatsoever of a structural failure.

Here is a link that might be useful: house collapse

The collapses make the news because they are NOT a very common event.

If they were like auto accidents that would barely merit a mention.