Rick Steves daypacks at museums

If this is a "mini" backpack, it is most likely fine. For example, Reina Sofia website says "Small handbags for carrying personal belongings and mini backpacks (˂ 30x30 cm) are allowed. If the galleries are accessed with mini backpacks, they must be worn at the front of the body to avoid disturbing other visitors and to ensure they do not brush up against the artworks"

Maybe pack a super lightweight tote and put your backpack in that when you enter the museum?
I love Baggu totes because they are light, washable and quite capacious!

I think the size of the bag is the issue. Unfortunately it is a safety thing, recall there were bombings at Atocha station years ago.

It isn't only the size. Lots of museums around the world have restrictions on backpacks because people are more likely to hit other people or the displays with them than with a shoulder bag. Some how human nature makes it harder for us to account for the extra size strapped on our back than when is it tucked under an arm.

A museum I frequent here offers visitors with backpacks bags in which to carry them such as Rebekah suggests.

I have worn the Rick Steves Civita daypack to most museums with no problem. Occasionally if it is very full I am asked to wear it in the front of me so as not to bump people or things. I have been asked a time one 2 to check it. For convenience at some museums I choose to check it as I recently did d’Orsay. . But most places I have no problem and can keep it with me.

I have seen small daypacks be an issue at museums. I believe it was Rjiksmuseum where we had to put all packs of any size in a rolling cargo cage. Then another museum in Rome made you store it yourself and the storage was across the street or some such. I carry a Nanopack backpack and don't have to deal with that really though. Whenever I go anywhere a backpack will be a problem, I take out the contents (water bottle, handsanitizer, chapstick, etc) out, tuck the Nanopack in its teeny sack, put it all in my pockets and go my merry way. Saved lots of time. Works even better when I have my jacket in my Nanopack because then I just put everything in my jacket pockets and wear or carry my jacket in the museum and don't get stuck carrying my jacket in my hand all day. I just put it back in the Nanopack when I don't want it anymore.

I don't know if they would waive you through with a tablet but probably.

I traveled with a guy who would fill his water bottle just over half, squeeze the air out, then screw the lid on and stick it in his back pocket. Security never said a word to him. I don't think they ever noticed.

Why I wear a cross-body messenger bag. It holds all the same stuff, is safe and secure, and museums are fine with it.
That said, most museums and cathedrals like in Burgos, have those lockers where you need a 1€ coin, which you get back when you open it. So, I do lock up my bag and jacket if those are available.

My small pack was not permitted in most places. My experience has been a bag check area where I check my bag ( I carry my valuables with me- camera and wallet) and it was rather nice to check my pack and not have to lug anything around.