If I sign a future-dated appointment letter at a company, can I still apply for other companies & decline to join this company later? [India, IT]

Question

I interviewed at a startup & cleared all rounds. Then the HR called me & said that they are willing to match my expected compensation & asked me to visit their office to sign the appointment letter:

HR: I can prepare the appointment letter for you and you can come down to our office for signature.

I haven't resigned from my current company yet.

I tried asking for an offer letter:

Me: I cannot sign the appointment letter right away. Please send me an offer letter first. I'll sign the appointment letter when I join you in 3 months.

To which they replied:

HR: We don't have an official offer letter. We usually send Letter of Intent which is confirmation of your selection and I can send you [compensation] breakup over email.

HR: We don't send an offer letter over email. If you want to go through it we can plan for the signing of Appointment letter at our office.

I feel something's not right here. It seems like they're trying to somehow force me to join their company & stop me from trying out for other companies.

I wish to be able to:

1. Use this company's offer as leverage to get better compensation in my current company.
2. Interview for other companies during my notice period & use this company's offer in compensation negotiations there too.
3. Not join this company if I get a better offer elsewhere.

So my questions are:

1. Should I insist on offer letter (& forgo this opportunity if they refuse to provide one), or will I be able to do all of the above if I sign the appointment letter right away?
2. Won't my current company & other companies ask to see this company's offer letter & the compensation breakup contained therein, before they can consider making me a better offer?
3. Would other companies refrain from making me an offer if I tell them that I've signed this company's appointment letter?

BTW, the following is how things played out 4 years ago when I joined my current employer:

1. Cleared interviews & received offer letter in email.
2. Accepted offer by replying on the email.
3. Signed both offer & appointment letter on the day of joining.

A couple of my colleagues are currently at step 2 above. They weren't asked to sign anything by their employers. One of them has accepted a better offer as well & plans to simply email the other company that she won't be joining them. I believe this is common practice in the Indian IT industry: candidates try to get multiple offer letters in hand, send email acceptance to all incoming offers, use competing offers to drive up their compensation, & eventually join just 1 company, informing the others over email, that they won't be joining them at all. Some employers even follow up with candidates, asking for a joining confirmation (after offer acceptance), but even if the candidate reconfirms their intention to join, they can back out of it later.

My primary concern is that if I sign the startup's appointment letter, will I get locked-in with them, unable to seek other opportunities, unable to use their offer as leverage in other compensation discussions I might have later on. I have a 3 months notice period, during which, I wish to apply to several companies & join just the best offering company eventually.

Answer 1

From the comments:

"Frankly, I don't wanna join this startup at all! The ONLY reason I want their offer is so I can use it as leverage to drive up my compensation in other companies. So I need a safe way of accepting their offer (without signing anything), that I can later turn down."

This is an important piece of context that should have been in the original post! You are concerned about them trying to lock you into a job that you don't really want; have you considered that they are concerned about you acting in bad faith too? If you were a hiring manager and had reasonable suspicion that the person in front of you was a time-waster, would you simply hand them something valuable that they wanted?

If you don't feel that you are getting fair compensation at your current workplace, then talk to your manager. Waving an offer letter from another company in their face has a risk of getting the response "go there then, as we're not going to match that under duress". And if you back down at that point, you're marked as both a flight risk and a bluffer.