Mom has a bed and will adjust the top and bottom of the bed, like a hospital bed. Is she going to need a hospital bed at some point?

I know this is overwhelming. So what I would do is get an order from her doctor concerning in home therapy. Let the physical therapist evaluate the situation. Have her talk to you only not the aide. She can ask the aide questions but she needs to talk to you about what she thinks will help Mom.

If the therapist thinks a hospital bed would be good for Mom then get a doctors order for it. Go to a store that sells durable equipment. Medicare may pay for the bed. Be aware though that Medicare is weird about if they have bought one thing they may not buy another. I oversaw a durable equipment loan closet. I had a client come in that knew what was what. So she rented what she knew if Medicare paid for they wouldn't pay for the bed she eventually would need.

Don't ask your caregiver's boss unless the boss is also a qualified occupational therapist. There are dozens of ways of helping a person to stand, and of transferring them even when they can't, and it's a question of getting the right kit for your mother. If the agency hasn't got one they can refer you to, try asking your local hospital or your mother's PCP.

And the right training, of course. Have you watched the caregiver supporting your mother? It's one way of learning how it's done :)

There is no comparison between a lift chair and a lift. The chair requires one to get into it. A lift is used FROM the bed to get into a chair. And yes, for someone debilitated to a large extent a hospital bed is the only answer. They raise up off the floor to the level at which the caregiver can change the patient. Are you soon looking at hospice care, do you know? They will often supply these items for the patient.
I advise strongly against a chair that lifts. It would be a waste of money at this point. Lifts and hospital bed mattresses are also washable, another plus.
Wishing you good luck. This isn't a matter of competence. It is a matter of caregivers being able to do this work without breaking their backs.

What size is mom's bed? If it is any larger than a Twin size it is difficult to roll a person from one side to the other in a larger bed.
Are there rails that mom can hold onto? These can be helpful not just when getting her up but when you roll her she can hold onto a rail.
Is the mattress washable? If she becomes incontinent it is easy to change the bedding and wipe down the mattress on a Hospital bed, not so easy with a mattress that has a cloth covering.
Also hospital bed mattresses are now typically with air flow or alternating pressure mattress to help prevent pressure sores.
Does her bed go up and down? Easier for someone to change or bathe a person in a bed if it is higher so that the caregiver's "body mechanics" work with them rather than trying to bend over.
Some of the things that your mom might need in the future.
Sit to Stand. This will help with transfers from the bed to a chair or commode. Great if your mom has some upper body strength and some leg strength.
Hoyer Lift. this will be what would be used if mom can not hold onto the Sit to Stand and has lost strength in upper body and legs.
There are different slings used with a Hoyer.
One with leg straps. this allows a more seated position.
Full sling.
And a full sling, usually mesh with a commode hole. This is great if you are getting mom into the shower. You can shower her with the sling on and easily lift her back to the bed to finish drying and dressing.
Any of these can be ordered by her doctor and usually covered by insurance.
If mom is on Hospice all of these would be provided by Hospice. As well as other supplies that you might need.
Do not feel incompetent. This is a whole 'nuther world that we get into as caregivers.
Also begin to think about the best placement for the bed. Is the bedroom actually the best place? Is mom isolated in the bedroom? If you moved the bed to a kitchen, dining room, living room area would that make mom feel more involved with the family?

If she is becoming so compromised her caregiver recommends a lift I don't think a lift chair is going to be a worthwhile investment, they can help people who are having a little difficulty getting up from a chair but the person needs to be able to retain their balance as the chair lifts and then be able to stand and walk independently - my mom's former caregiver had many stories of people who had been dumped on the floor by these chairs. When my mom got to this point I looked to the future and had her fitted with customized a tilt in place wheelchair.
As for the hospital bed - I would question exactly what features she is looking for, although it is easier for caregivers to raise or lower the height of the bed when transferring or changing their patient most home caregivers learn to do just fine without having one.
My advice to you would be to have an OT and/or PT come into your home to evaluate your mother's abilities and make recommendations about what kind of devices you need, they would also know about any benefits available to help with the cost and how to apply for them.