Will a 6x10 glue-lam be suitable to replace a 4x10 porch beam with one fewer post?
Yes, using a 5 1/8” x 10 1/2” glulam beam will support a 16’ long header. You don’t give all the loading information required, but I’ll make some assumptions and you tell me if I’m wrong.
Loading from your covered patio is 20 psf snow live load plus 10 psf dead load (lumber, roofing, etc.) for a total load of 30 psf. Because the covered patio is 12’ deep, then 6’ bears on the header... so, 6’ x 30 psf = 180 plf along the header.
The 5 1/8” x 10 1/2” glulam will support 600 plf. Therefore, 600 plf - 180 plf = 420 plf allowed from the “house side” on the header. So if the span onto the header is less than 28’ then the glulam will support the load. (Remember, 14’ will rest on the new header and the other 14’ will rest on the other side of the house.)
If you live in a climate that has more than 20 psf of snow, the header will still work if snow lasts less than 90 days, due to an impact allowance.
Also, you may use a 4x4 for support at each end in lieu of the 4x6 you’re suggesting. Bearing value on each 4x4 is 600 lbs. x 16’/2 = 4800 lbs. So actual load is 4800 lbs. / 3.5” x 3.5” = 391 psi. ~ 385 psi allowable.
Make sure the beam and posts are connected together with a metal connector, such as Simpson ACE.
Likewise, connect the post base to the sole plate with a metal connector.
The last issue to check is bearing on footings. The code allows 1500 psf soil bearing value without a geotechnical report. Therefore, the maximum loading on the soil is 4800 lbs. plus 600 lbs. of concrete for a total load of 5,400 lbs.  If your footings are a minimum of 16” wide, then you’d need only the perimeter strip footing to be a minimum of 30” long.
When you order your glulam, be sure to order 2400 f. (Not 2200 f.)