is chivalry really patriarchy?

In the Netflix show Loudermilk, the main character (a man in his forties), holds the café door open for a young attractive woman, then complains loudly when, now ahead of him in line, she gives a long, complicated coffee order.
“I held the door for you!” he gripes.
“Yes, but just because I’m hot.” she replies.
 
Is chivalry just another word for patriarchy? Pulling out the chair, holding the door, carrying the bag…is there not a suggestion of weakness, fragility, disability? “That looks heavy, let me get that for you.” Insert optional bicep flex. Or, as insinuated in the scene above, is it all about “If I do this, I’ve got a chance?” Either way, I don’t like the implications. Of course, if my arms are full, and someone holds the door, that’s great. But shouldn’t we do that for everyone, regardless of gender? Isn’t everyone equal?

Chivalry developed as a code of conduct to teach knights how to behave, since apparently they’d spend all their time raping and plundering otherwise. Surely we’ve moved beyond that now? Even the clichéd chivalrous gesture of laying down a cloak over a mud puddle, so the woman won’t get dirty, annoys me for two reasons. One, why was it so unthinkable that the woman might get dirty? At a time without roads, showers, or Tide? Wasn’t everyone pretty much covered in dirt all the time? And two, who’s going to wash that now-muddy cloak? Mr. Chivalrous? I doubt it.
 
I’ve spoken to men before about offering to help, rather than swooping in to yank things out of a woman’s hand, or do a task for her. Most women, given the opportunity, can do things for ourselves, and I’d like a chance to put my own suitcase into the vehicle, carry the bag of dog food, and use the drill, axe and chainsaw. I’m just as likely to ask my tall sister-in-law to help me reach something on a high shelf as my gargantuan nephew; it’s just that I’m too lazy to get a chair, not that somehow the Y-chromosome is the saviour.
 
We all need help sometimes and it should be given, without judgement, by anyone available. Even if I can’t push your car out of the snow, I can dig, I can call CAA. Even if the elderly person with a walker behind you is a man, you should still hold the door open. Yes, you should give your seat on the bus to the pregnant woman, but also the obviously exhausted male night shift worker. Can one give preferred treatment (women served first, seated first, etc.) and not come across as condescending and patriarchal? Maybe some women like this status, but I’d prefer that we all be the same.
 
Some ideas, for those who are feeling chivalrous:
1) When doing a “chivalrous” act, consider your motivation (hopefully not the same as Loudermilk).
2) When wanting to help, consider saying “Would you like some help?” Let the other person decide if they require your assistance. Don’t assume that only men can/should do certain tasks.
3) Do nice things for everyone, all day. Open doors, pull out chairs, knock yourself out. But do it equally.
In my opinion, these three things would make a really big difference.